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Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes
Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes Guilherme Augusto Lopes da Silva, Rodrigo Nicoletti n University of São Paulo, São Carlos School of Engineering, Dept. of Mechanical Engineering, Trabalhador São-Carlense 400, 13566-590 São Carlos, Brazil
a r t i c l e i n f o
abstract
Article history: Received 1 June 2016 Received in revised form 10 February 2017 Accepted 24 February 2017 Handling Editor: M.P. Cartmell
This work focuses on the placement of natural frequencies of beams to desired frequency regions. More specifically, we investigate the effects of combining mode shapes to shape a beam to change its natural frequencies, both numerically and experimentally. First, we present a parametric analysis of a shaped beam and we analyze the resultant effects for different boundary conditions and mode shapes. Second, we present an optimization procedure to find the optimum shape of the beam for desired natural frequencies. In this case, we adopt the Nelder-Mead simplex search method, which allows a broad search of the optimum shape in the solution domain. Finally, the obtained results are verified experimentally for a clamped-clamped beam in three different optimization runs. Results show that the method is effective in placing natural frequencies at desired values (experimental results lie within a 10% error to the expected theoretical ones). However, the beam must be axially constrained to have the natural frequencies changed. & 2017 Published by Elsevier Ltd.
Keywords: Natural frequency Optimization Embossed pattern Finite element method Mechanical vibration Slender beams
1. Introduction Manufactured products can present excessive vibration during operation due to the inherent dynamics of their structure and due to the source of excitation. Structure dynamics affects vibration depending on the location of resonance frequencies in the frequency range of operation, on the damping level of the structure, on the appearance of non-linear modes during operation, and on the presence of vibro-acoustic coupling. The source of excitation affects vibration depending on the amplitude level of the load, which can lead the system to high vibration levels even in low frequency response regions of the structure. Unless the Engineer has control of the excitation source, the problem of reducing the vibratory response of a system lies on the modification of the structure, i.e. on the modification of its dynamics. The dynamics of a structure can be modified either actively or passively. An active modification of structure dynamics implies the adoption of sensors and actuators in a control loop feedback. In this case, the fine-tuning of the controller's gains moves the poles and zeros of the structure to new positions in the complex plane, thus altering the structure dynamic characteristics in closed loop. The passive modification refers to a physical change of the structure in terms of its geometry and/or its material, thus affecting the inherent mass and stiffness of the system and, consequently, affecting its dynamic characteristics. In both cases, we can shift natural frequencies, placing them at desired values to avoid operating frequency regions or to create frequency dead zones in the structure. n
Corresponding author. E-mail addresses: [email protected] (G.A.L.d. Silva), [email protected] (R. Nicoletti).
http://dx.doi.org/10.1016/j.jsv.2017.02.053 0022-460X/& 2017 Published by Elsevier Ltd.
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Nomenclature Ci E desired fi k fi nd fi F h K L M
i-th weighing factor (–) material Young modulus (N.m −2) i-th desired natural frequency (Hz) i-th natural frequency at k-th iteration (Hz) i-th natural frequency of non-deformed structure (Hz) objective function (–) height of cross section (m) stiffness matrix beam length (m) inertia matrix
T ui ^ u i u, v w x
α δmax ρ θ φ ^ φ
transformation matrix i-th eigenvector (–) i-th normalized eigenvector (–) nodal displacements (m) width of cross section (m) vector of degrees of freedom beam finite element tilting angle (rad) maximum deformation of the beam (m) material density (kg.m −3) nodal angular displacement (rad) shape of the beam (–) normalized shape of the beam (–)
In this work, we will focus on the placement of natural frequencies of the structure to desired frequency regions. More specifically, we will focus on the placement of natural frequencies of a beam by changing its shape (passive modification of the structure). The optimization of beams subjected to frequency restriction has been studied since the late 1960s [1–3]. However, such studies focused on parameter optimization (not shape optimization). The actual idea of optimizing the shape of beams and plates aiming at desired eigenfrequencies started in the 1990s by changing the local thickness of the beam or plate [4,5]. For that, the authors defined the thicknesses of different regions of the structure as design variables of the optimization procedure. Finite element modeling was naturally suitable for the optimization procedure because this numerical method divides the structure in discrete regions of constant thickness. Experimental results showed the effectiveness of the methodology in placing the natural frequencies of the structure at desired specific values [6,7]. The drawback of changing the thickness of the structure lies on practical difficulties of manufacturing. In this case, machining is a cumbersome process and stamping (a more feasible solution) requires a careful design of the dies. Alternatively, we can change the geometry of the beam or plate considering constant thickness. For example, the coordinates of points in the boundary of the structure can be used as design variables in the optimization procedure [8]. Hence, the geometry changes in the plane defined by the larger dimensions of the beam or plate (thickness is perpendicular to this plane and it is kept constant). Again, we can shift natural frequencies of the structure towards higher values, but this procedure cannot be applied to structures with fixed boundaries (e.g. welded or bolted plates at their whole perimeter). If the boundaries are fixed but the plate has holes whose geometry can be changed, we can optimize the shape of the holes instead, as numerically shown in [9]. In this case, one reached a 20% increase of the first natural frequency and a 12% increase of the fourth natural frequency of the plate in two different optimization runs. We can also design optimal reinforcements to the structure with constant thickness, where the structure geometry remains the same. In [10], the adopted design variables were the length, orientation, and coordinates of the reinforcement to be applied perpendicular to the plane of a plate. In this case, the optimization procedure was applied to find the best position and length of the reinforcement that maximizes the first natural frequency of the plate. The obtained results showed an increase of 10% of the first natural frequency of the plate. Other examples of optimal positioning of reinforcements towards structure dynamic modification are found in [11,12]. These works deal with reinforcements with constant height, but it is also possible to optimize the shape of the reinforcements. In [13], the position and length of the reinforcements are known and fixed, and the shape of the reinforcements (height) is optimized towards maximization of the first natural frequency of the plate. The results showed an increase of near 40% of the first natural frequency of the plate in study. Another way of changing the inherent dynamics of beams and plates with constant thickness is through embossing patterns on them. In this case, the structure is locally deformed to increase stiffness and, consequently, the natural frequencies. In [14], an optimization procedure finds the best locations of grooves in the surface of a plate. The results show an improve of ∼270% of the first natural frequency, and the final deformed shape of the plate (optimum shape) is very suitable for sheet metal stamping. In [15] and [16], the grooves are defined by the mode shapes of the structure in original condition (not deformed), and the final deformed shape of the structure is a linear combination of these mode shapes. The authors used a three-level full factorial design of experiments (DOE) to find the weighing factors of the linear combination of mode shapes that shifted the natural frequencies towards the desired values. Such DOE approach is suitable for time consuming computational models, with high number of degrees-of-freedom. The present work gives an additional contribution to the embossed pattern methodology presented in [15] and [16]. Here, we further investigate the effects of combining mode shapes to shape a beam to change its natural frequencies, both numerically and experimentally. First, we analyze the resultant effects for different boundary conditions and mode shapes. This analysis allows a better insight into the phenomenology of the problem. Second, we present an optimization procedure to find the optimum shape of the beam for desired natural frequencies that is different from the one used in [16]. In this case, we adopt the Nelder-Mead simplex search method, which allows a much broader search of the optimum shape in the Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 1. Degrees-of-freedom of the finite element used to model the beam.
solution domain than the design of experiments adopted by [16], for models whose tests are not time consuming (cheap and quick to calculate). Finally, the obtained results are verified experimentally for a clamped-clamped beam. We performed three different optimization runs and the results show that the experimental natural frequencies of the shaped beam lie within a 10% error to the expected theoretical ones.
2. Mathematical modeling of the shaped beam This work focuses on the natural frequencies of permanently deformed beams (shaped beams) and, on the optimum deformation (shape) of the beam that results in desired natural frequency values. Hence, the beams in analysis are not straight and they have a certain known geometry. We will neglect in the mathematical modeling any residual stress in the beam due to the plastic deformation process (it is assumed that the beam went through a stress relief process). The mathematical modeling of the beam is based on Euler-Bernoulli finite elements with axial nodal displacements [17]. The axial nodal displacements are important to account for the coupling effects that arise when the elements of the model present a relative angle between each other. Hence, the vector of degrees-of-freedom of the finite element in local coordinates is:
x e = { u1 v1 θ1 u2
v2
θ2}
(1)
where ui is the in-plane (axial) displacement of the i-th node of the element, vi is the out-of-plane (perpendicular) displacement of the i-th node of the element, and θi is the angular displacement of the i-th node of the element (Fig. 1). In the case of a non-straight beam (shaped beam), the beam is modeled by straight elements with a relative angle between each other to follow the shape of the beam. Hence, the elements of the model will present a tilting angle α in relation to the global coordinate system (Fig. 2). Hence, we must perform a coordinate transformation to obtain the global matrices of the model in global coordinates (X,Y):
MGe = T T MeT
(2)
K Ge = T T K eT
(3)
MGe
K Ge
where and are the global inertia and stiffness matrices of the element, Me and K e are the inertia and stiffness matrices of the element in local coordinates, and T is the transformation matrix (all these matrices are found in [17]). By arranging the global matrices of the elements, we derive the equations of motion of the beam:
(4)
Mx¨ + Kx = 0
where M is the inertia matrix of the beam model, K is the stiffness matrix of the beam model, x¨ is the acceleration vector, and x is the displacement vector. We will not include damping in the model for simplicity. This assumption is acceptable for slender beams, which usually present low damping factors (below 5%). It is also important to emphasize that the adopted mathematical modeling, based
Fig. 2. Example of shaped beam: (a) original geometry, (b) finite element model.
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Table 1 Boundary conditions of the beam adopted in the analysis. Free-free (FF)
Cantilever (CB)
Simply supported axial free (SSF)
Simply supported (SS)
Clamped-clamped (CC)
Table 2 Properties of the baseline beam in the analyses. Length (L)
0.600
m
Width (w) Height (h) Density (ρ) Young modulus (E)
0.030 0.003 2,688 69 109
m m kg.m 3 N.m 2
on Euler-Bernoulli finite elements, are only valid for slender beams. Thus, considering Eq. (4), we can solve the associated eigenvalue problem and find the eigenvalues (λi) and eigenvectors (u i ) of the system. The adopted boundary conditions are listed in Table 1 and the properties of the baseline beam are listed in Table 2. The baseline beam has a rectangular cross section and it has a height-to-width ratio (h/w ) of 0.1 and a height-to-length ratio (h/L ) of 0.005. The number of finite elements in the model of the beam affects the resultant eigenvalues (natural frequencies) and, a convergence analysis was performed. In this case, there was no significant variation of the first five natural frequencies of the beam model when adopting 36 finite elements or more, irrespective of the adopted boundary condition. Hence, we will adopt 36 finite elements in the model in the following analyzes.
3. Parametric analysis of the deformed beam In this section, we will change the shape of the baseline beam and analyze the effects on the resultant natural frequencies, using the mathematical model described before. For that, we will use the eigenvectors (mode shapes) of the baseline beam (straight beam) as shape functions. That means, the beam will take the geometry of a given mode shape of the baseline beam, depending on the adopted boundary condition. Considering that an eigenvector only gives relative displacements, we must define the scale with which we will shape the beam. This scale is defined by the maximum deformation of the shaped beam (δmax in Fig. 3). In this case, the beam is
Fig. 3. Maximum deflection of the shaped beam in relation to the baseline beam.
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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shaped following the chosen eigenvector in a proportion that the maximum distance between the shaped beam and the baseline beam is δmax. Thus, by changing the value of δmax, the geometry of the whole beam is changed proportionally to the chosen eigenvector (mode shape). The maximum deformation of the shaped beam (δmax) gives the information of how much deformed is the beam. In the following analyses, we will adopt a different parameter to inform the level of deformation of the shaped beam: the maximum relative deformation. The maximum relative deformation is the ratio between δmax and the height of the beam cross section (h). Hence, if the maximum relative deformation is 1.0, it means that the maximum distance between the shaped beam and the baseline beam is one height of the beam cross section. 3.1. Effect of boundary condition Here, the baseline beam will be shaped by its first mode shape, depending on the adopted boundary condition. The maximum relative deformation of the beam varied from 0 (straight beam) to 3.0 (maximum distance between the shaped beam and the baseline beam is three heights of the beam cross section). The results are presented in Fig. 4. As we can see, in the cases of free-free, cantilever and, simply supported axial free boundary conditions (Fig. 4(a) to (c)), there is no significant variation of the first four natural frequencies of the shaped beam in relation to those of the baseline beam (straight beam). In the case of the simply supported boundary condition (Fig. 4(d)), the value of the first natural frequency of the beam significantly increases as the maximum relative deformation increases. For maximum relative deformations above 1.5, there is a shift in the order of the mode shapes (the first mode shape becomes the second mode shape, and vice-versa). In the case of the clamped-clamped boundary condition (Fig. 4(e)), not only the first natural frequency increases as the maximum relative deformation increases, but also the third and fifth natural frequencies present some variation towards higher values. The order of the modes changes for maximum relative deformations above 2.5. The results in Fig. 4 show that the shaped beam must be axially constrained to present variations in its natural frequencies. In addition, by shaping the beam in its first mode shape geometry, we affect the first natural frequency predominantly. In the case of the clamped-clamped beam, we also see effects on the third and fifth natural frequencies, whose mode shapes are from the same family of the first mode shape (even functions). 3.2. Effect of mode shape geometry Considering that, the first natural frequency is predominantly affected by shaping the beam in its first mode shape, we adopt other mode shapes to verify their effect on the natural frequencies. Figs. 5(a) and 5(b) present the first five natural frequencies of the baseline beam shaped by its second mode shape geometry, depending on the adopted boundary condition. The shaped beams not axially constrained (free-free, cantilever, simply supported axial free) presented again no variation in their natural frequencies and the results are omitted. In the case of the simply supported boundary condition (Fig. 5(a)), the value of the second natural frequency of the beam significantly increases as the maximum relative deformation increases. The order of the mode shapes also changes as the value of the natural frequency associated to the second mode shape surpasses the succeeding natural frequencies. The value of the first natural frequency is not affected, as well as the values of all other natural frequencies analyzed. In the case of the clamped-clamped boundary condition (Fig. 5(b)), not only the second natural frequency increases as the maximum relative deformation increases, but also the fourth natural frequency present some variation towards higher values. The order of the modes changes as the maximum relative deformation increases. The value of the first natural frequency is also not affected, as well as the values of all other odd natural frequencies analyzed (third and fifth). Figs. 5(c) and 5(d) present the first six natural frequencies of the baseline beam shaped by its third mode shape geometry, depending on the adopted boundary condition. The shaped beams not axially constrained (free-free, cantilever, simply supported axial free) presented again no variation in their natural frequencies and the results are omitted. In the case of the simply supported boundary condition (Fig. 5(c)), the value of the third natural frequency of the beam significantly increases as the maximum relative deformation increases. Again, the order of the mode shapes changes as the value of the natural frequency associated to the third mode shape surpasses the succeeding natural frequencies. The values of the first and second natural frequencies are not affected, as well as the values of all other natural frequencies analyzed. In the case of the clamped-clamped boundary condition (Fig. 5(d)), not only the third natural frequency increases as the maximum relative deformation increases, but also the fifth natural frequency present some variation towards higher values. The order of the modes changes as the maximum relative deformation increases. The values of the first and second natural frequencies are not affected, as well as the values of all other even natural frequencies analyzed (fourth and sixth). The results in Fig. 5 show that by shaping the beam in its i-th mode shape geometry, we predominantly affect the i-th natural frequency. The natural frequencies of the baseline beam whose values are below this i-th natural frequency remain unaffected. In the case of the clamped-clamped boundary condition, we also see the effects on the natural frequencies whose mode shapes are from the same family of the i-th mode shape. In this case, frequency veering is observed when the frequency of the i-th mode reaches the frequency of a mode of the same family. Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 4. Natural frequencies of the shaped beam (first mode shape geometry) as a function of the maximum relative deformation: (a) free-free, (b) cantilever, (c) simply supported axial free, (d) simply supported, (e) clamped-clamped.
4. Optimization procedure The results presented in the previous section show that it is possible to increase a natural frequency of the beam by shaping it with the geometry of a mode shape of the beam in its original condition (straight beam). The question that follows is whether two or more mode shapes can be combined to change more natural frequencies of the beam simultaneously. The answer is yes, as demonstrated in [15,16]. Actually, the problem is how to combine the mode shapes to find the Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 5. Natural frequencies of the shaped beam as a function of the maximum relative deformation: (a) simply supported (2nd. mode geometry), (b) clamped-clamped (2nd. mode geometry), (c) simply supported (3rd. mode geometry), (d) clamped-clamped (3rd. mode geometry).
geometry of the beam that results in desired natural frequencies. In [16], the authors did a linear combination of the mode shapes, whose coefficients (weighting factors of the linear combination) were obtained by Design of Experiments, an exhaustive try out process. Here, we present an optimization procedure to find these coefficients of the linear combination of mode shapes. Consider the mathematical model of the beam (Eq. (4)). We can solve the associated eigenvalue problem and find the eigenvalues (λi) and eigenvectors (u i ) of the system. Considering that the eigenvectors are linearly independent vectors, we can use them as a basis to find the optimized shape of the beam, in the form: N
φ=
∑ Ciui i=1
(5)
where C i is the weighing factor of the i-th mode shape of the basis, N is the number of mode shapes that form the basis, and φ is the final geometry of the shaped beam. The task of the optimization procedure is to find appropriated values for the weighing factors C i that make the beam, shaped in the geometry φ, present the desired natural frequencies. The problem with Eq. (5) is that eigenvectors represent mathematically a direction in space [18]. Hence, the elements of the eigenvectors have no absolute values, and we must adopt a norm to represent them. In the present case, the elements of the eigenvectors are normalized by the respective maximum absolute value:
^ = u i
ui max (|ui |)
(6)
^ is the i-th normalized eigenvector of the system. Thus, the normalized eigenvectors will always present element where u i values between -1 and 1. Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Another problem with Eq. (5) is that the function φ can suffer from scaling and, consequently, it can grow indefinitely for the same linear combination of eigenvectors. If the weighing factors Ci are scaled by a scalar a, the resultant shape will be:
φ* = aφ
(7)
where φ* is a scaled version of the shape φ. If a tends to infinity, the deformed shape will also tend to infinity. However, both deformed shapes represent the same linear combination of eigenvectors (same family of shapes). To avoid such infinite number of similar solutions to the problem, we also normalize the shape by the respective maximum absolute value: N
φ=
∑ Ciu^ i
(8)
i=1
^=δ φ max
φ max (|φ|)
(9)
where δmax is the maximum deformation of the beam (Fig. 3). Thus, the normalized shape will present element values between −δmax and δmax. The maximum deformation of the beam is defined previously to the optimization procedure (parameter of design). ^ , we can find the respective vector of tilting angles of the finite element model After calculating the normalized shape φ (vector α ). The finite element model of the beam is then updated and the associated eigenvalue problem will result in new eigenvalues and eigenvectors of the system. In this case, the new eigenvalues (natural frequencies) will refer to the shaped ^. beam in the geometry given by φ Hence, we can write an optimization algorithm whose flowchart is shown in Fig. 6. Initially, we solve the eigenvalue problem of the system in original condition (straight beam) and we obtain the eigenvectors of the system. These eigenvectors are normalized according to Eq. (6) and they form the basis for the shape optimization. Given an initial set of weighing factors, the looping of optimization starts. We calculate the new geometry of the beam with Eqs. (8) and (9). The finite element model of the beam is updated with the new geometry and, the eigenvalue problem is solved to find the natural frequencies of the shaped beam. We use the natural frequencies of the shaped beam with the new geometry to calculate the objective function F.
Fig. 6. Flowchart of the optimization algorithm.
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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In this work, we adopt three different objective functions to perform the shape optimization of the beam. If the objective is the maximization of the first natural frequency of the beam, then we must minimize the objective function:
f 1nd
F (k ) =
f1k
(10)
k f1
where is the first natural frequency of the shaped beam calculated in the k-th iteration of the algorithm, and f 1nd is the first natural frequency of the baseline beam (straight beam). If the objective is the maximization of the j-th natural frequency of the structure keeping the other natural frequencies unchanged, then we must minimize the objective function: j−1 ⎛
F (k ) =
∑ ⎜⎜ 1 − i=1
⎝
2
f nd f ind ⎞ ⎟ + j + ⎟ fik ⎠ f kj
2 ⎛ f nd ⎞ ⎜1 − i ⎟ ⎜ ⎟ fik ⎠ i=j+1 ⎝ N
∑
(11)
k fi
nd fi
where is the i-th natural frequency of the shaped beam calculated in the k-th iteration of the algorithm, and is the i-th natural frequency of the baseline beam (straight beam). If the objective is the positioning of natural frequencies at specific desired values, then we must minimize the objective function:
F (k ) =
N
⎛
i=1
⎝
∑ ⎜⎜ 1 −
2
f idesired ⎞ ⎟ ⎟ fik ⎠
(12)
desired fi
where is the desired value for the i-th natural frequency of the shaped beam. After calculating the value of the objective function (Fig. 6), we implement the algorithm fminsearch of software MATLAB to find new values of the weighing factors. The algorithm fminsearch performs an unconstrained nonlinear optimization (minimization of the objective function) using the Nelder-Mead simplex search method [19]. This is a direct search method that does not use numerical or analytical gradients. In addition, it may only give local optimum solutions. The adopted convergence criteria are the tolerance value of the objective function (10 7 in the present work), and the tolerance value of the design variables, i.e. the tolerance value of the weighing factors (10 5 in the present work). The adopted stop criterion is the maximum number of iterations (300 iterations in the present work).
5. Experimental results We performed the optimization analysis in an aluminum beam whose properties are listed in Table 3. Initially, we determined experimentally the natural frequencies and mode shapes of the beam under a clamped-clamped boundary condition. For that, the beam (Fig. 7–1) was mounted on fixed clamping blocks (Fig. 7–2), and a electrodynamic shaker (Fig. 7–3) was connected to the beam 100 mm apart from one of the clamped ends. A load cell mounted in the shaker side of the stinger (Fig. 7–4) measured the excitation force. The vibrating response of the beam under excitation was measured by a laser vibrometer (Fig. 7–5) at every 50 mm of the beam (11 equally spaced measuring points). The excitation signal used in the experiment was a chirp signal, ranging from 10 to 400 Hz and back to 10 Hz in 8 s (linear variation). This signal was repeated 10 times, thus resulting in a total acquisition period of 80 s. The adopted acquisition rate was 5 kHz. The frequency response functions were calculated by the H1 and H2 estimators [20], with Hanning window of 5,000 points and overlap of 3,500 points. Table 4 presents the obtained natural frequencies and associated mode shapes of the beam in the frequency range of study (below 400 Hz). The measurement error of 0.5 Hz represents the measurement accuracy of the system. We built the mathematical model of the beam using the Euler-Bernoulli beam finite elements described in Section 2. We adopted 36 finite elements to represent the beam under clamped-clamped boundary condition. The beam was in its original condition (straight beam) and, consequently, all finite elements had null tilting angle (α ¼ 0). The parameter values used in the model were the same as those presented in Table 3. By mounting the global inertia and stiffness matrices and solving the Table 3 Properties of the aluminum beam in analysis. Property
Value
Unit
Total length Width (b) Thickness (h) Material density
0.6 31 3 2688
m mm mm
Young modulus (E)
kg.m 9
69 ×10
N.m
−3
−2
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 7. Test rig of the clamped-clamped beam: (1) beam, (2) clamping blocks, (3) shaker, (4) load cell, (5) laser vibrometer.
Table 4 Experimental natural frequencies of the aluminum beam under clamped-clamped boundary condition.
1 2 3 3
Natural frequency (Hz)
Mode shape
42.57 0.5 107.5 7 0.5 195.5 7 0.5 340.57 0.5
First bending Second bending Third bending Fourth bending
Table 5 Comparison between experimental and numerical natural frequencies of the beam (straight beam).
1 2 3 3
Numerical (Hz)
Experimental (Hz)
Error (%)
42.14 105.22 195.73 339.68
42.5 70.5 107.5 7 0.5 195.5 7 0.5 340.5 70.5
0.8 2.1 þ 0.1 0.2
eigenvalue problem of the system, we obtained the numerical results shown in Table 5. These results were obtained after including an additional lumped mass of 40 g in the degrees-of-freedom of the model related to the position of the attached shaker (node located 100 mm apart from the clamped edge). Such addition of a lumped mass in the model accounted for the stinger attached to the structure. We observe that the numerical results of natural frequencies present good correlation to the experimental ones. Such good correlation can also be noted in the resulting mode shapes (Fig. 8). These mode shapes presented in Fig. 8 form the basis of eigenvectors in the optimization procedure applied to the present case (clamped-clamped beam). Considering that the mathematical model is correlated to the real structure, it was used in the optimization algorithm. Here, we performed three different optimization analyzes:
maximization of the first natural frequency, where the objective was to increase the global stiffness of the structure; maximization of the third natural frequency, keeping the first and the second natural frequencies unchanged, where the objective was to create a gap of resonances in the structure (dead zone);
positioning of natural frequencies, where the objective was placing the four natural frequencies at desired values. In all these analyzes, the maximum deformation of the shaped beam was a parameter previously defined. Therefore, the results are presented as a function of the maximum deformation of the beam (δmax). In addition, the initial guess of the weighing factors was null in all cases (Ci(0) = 0, ∀ i ). All four eigenvectors were used in the shape optimization (N ¼4), irrespective of the number of natural frequencies involved in the objective function. 5.1. Maximization of the first natural frequency The optimization procedure to maximize the first natural frequency of the beam adopted the objective function of Eq. (10), where f 1nd = 42.14 Hz (first natural frequency of the straight beam). The obtained numerical results are presented in Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 8. Comparison between the experimental and numerical mode shapes of the beam (straight beam): (a) first mode, (b) second mode, (c) third mode, (d) fourth mode.
Fig. 9. Natural frequencies of the optimum shaped beam as a function of the maximum relative deformation. (a) evolution of natural frequencies; (b) ratio of the objective function and initial objective function.
Fig. 9. As we can see, the optimum values depend on the maximum relative deformation of the beam. The first natural frequency increases as the maximum relative deformation increases, thus showing a similar behavior to what has been shown in the parametric analysis (Section 3). However, when the first natural frequency reaches the value of the second natural frequency, there is no significant change in the results, irrespective of the maximum relative deformation adopted. In this case, the first natural frequency of the beam remains the same for any maximum relative deformation above 2, and the value of the objective function remains constant (Fig. 9(b)). Considering these results, we manufactured an aluminum beam shaped with the optimum geometry with maximum deformation of 10 mm (max. relative deformation of 3.3). The optimum geometry is shown in Fig. 10, which is the result of a linear combination of the eigenvectors with the optimized weighing factors: C1 = 1.0000, C2 = 0.0346, C3 = 0.0983 and, C4 = 0.1292. The obtained shape of the beam was measured with a coordinate measuring machine and the results are shown in Fig. 10(a) whose measurement errors are70.1 mm. Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 10. Beam shaped with the optimum geometry for maximum deformation of 10 mm. (a) optimum shape; (b) shaped aluminum beam.
Table 6 Experimental natural frequencies of the beam shaped with the optimum geometry with maximum deformation of 10 mm. Comparison to the expected theoretical values.
1 2 3 3
Experimental (Hz)
Numerical (Hz)
Error (%)
98.5 7 0.5 107.5 7 0.5 265.0 7 1.0 368.57 0.5
105.23 105.24 261.62 403.97
-6.4 þ 2.3 þ 1.3 -8.8
By testing experimentally the beam shaped with the optimum geometry, following the same experimental procedure described in Section 5, we obtained the results presented in Table 6. As we can see, the error between the expected theoretical values and the real experimental values remained below 9%. This shows the effectiveness of the method to find the optimized shape of the structure that maximizes its first natural frequency. Considering that the mass of the structure remains unchanged, the stiffness of the structure is maximized. Looking at the resulting optimum geometry (Fig. 10) and at the optimized weighing factors, we can see that there is a preponderance of the first mode shape in the solution. Using the first eigenvector only as the geometry of the beam, we obtained the results shown in Fig. 11. As we can see, the results are very similar to those obtained in the optimization procedure using the complete basis of four eigenvectors (Fig. 9). This shows that, shaping the beam with the geometry of its first mode shape is as efficient as using a linear combination of the first four mode shapes in the maximization of the first natural frequency of the structure. In fact, as already demonstrated in Section 3, when the structure has the geometry of a given mode shape, we maximize the associated natural frequency of this mode shape. Hence, to maximize the first natural frequency, we can shape the beam with the geometry of the first mode shape of a straight beam. This represents an educated guess without performing the optimization analysis. However, it is important to emphasize that, there will be no control of the remainder natural frequencies, and they can change freely towards higher values.
Fig. 11. Natural frequencies of the shaped beam as a function of the maximum relative deformation (geometry of the first eigenvector).
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5.2. Maximization of the third natural frequency The optimization procedure to maximize the third natural frequency of the beam, keeping the first and the second natural frequencies unchanged, adopted the objective function of Eq. (11), where f 1nd = 42.14 Hz , f 2nd = 105.22, and f 3nd = 195.73 (first, second, and third natural frequencies of the correlated model). The obtained numerical results are presented in Fig. 12. Again, the optimum results depend on the maximum relative deformation of the beam. The third natural frequency value increases as the maximum relative deformation increases, in a similar way to what has been shown in Section 3, where only one eigenvector is used as the geometry of the shaped beam (third eigenvector). In this case, the third natural frequency of the beam remains the same for any maximum relative deformation of the optimum shape above 1. Consequently, the value of the objective function remains constant above this threshold value (Fig. 12(b)). Considering these results, we manufactured an aluminum beam shaped with the optimum geometry for a maximum deformation of 4 mm (max. relative deformation of 1.3). The optimum geometry is shown in Fig. 13, which is the result of a linear combination of the eigenvectors with the optimized weighing factors: C1 = 0.9140, C2 = 0.2906, C3 = 1.0000 and, C4 = 0.0888. The obtained shape of the beam was measured with a coordinate measuring machine and the results are shown in Fig. 13(a) whose measurement errors are70.1 mm.
Fig. 12. Natural frequencies of the optimum shaped beam as a function of the maximum relative deformation. (a) evolution of natural frequencies; (b) ratio of the objective function and initial objective function.
Fig. 13. Beam shaped with the optimum geometry for maximum deformation of 4 mm. (a) optimum shape; (b) shaped aluminum beam.
Table 7 Experimental natural frequencies of the beam shaped with the optimum geometry for maximum deformation of 4 mm. Comparison to the expected theoretical values.
1 2 3 3
Experimental (Hz)
Numerical (Hz)
Error (%)
43.5 7 0.5 106.57 0.5 308.5 7 0.5 347.8 7 0.5
42.17 105.21 339.70 339.71
þ3.2 þ1.2 -9.2 þ2.3
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 14. Natural frequencies of the shaped beam as a function of the maximum relative deformation (geometry of the third eigenvector).
By testing experimentally the beam shaped with the optimum geometry, following the same experimental procedure described in Section 5, we obtained the results presented in Table 7. As we can see, the error between the expected theoretical values and the real experimental values remained below 10%. Looking at the optimum geometry (Fig. 13) and at the optimum weighing factors, we can see that there is a preponderance of the first and third mode shapes in the solution. However, the results of natural frequencies were very similar to those using the third mode only (Fig. 14). Again, the third mode shape of the straight beam represents an educated guess to maximize the third natural frequency of the structure, keeping the first and second natural frequencies unchanged (no need of an optimization analysis). In both cases (shape geometry composed of a full set of eigenvectors or of the third eigenvector only), we obtained a frequency range of ∼200 Hz without any resonances in the beam (from ∼106 Hz to ∼310 Hz), thus showing the effectiveness of the method. 5.3. Positioning of natural frequencies The optimization procedure for positioning the natural frequencies of the beam at desired values adopted the objective function of Eq. (12). The desired values are 50, 150, 250, and 350 Hz, which means that all natural frequencies should be changed. Fig. 15 presents the optimum results as a function of the maximum relative deformation of the beam. As we can see, the optimization algorithm achieves the objective for maximum relative deformations between 1 and 2.3 (Fig. 15(b)). Above the value of 2.7, there are some oscillations in the solution, thus indicating that there are regions of optimality depending on the maximum relative deformation of the beam. Considering these results, we manufactured an aluminum beam shaped with the optimum geometry for a maximum deformation of 4 mm (max. relative deformation of 1.3). The optimum geometry is shown in Fig. 16, which is the result of a linear combination of the eigenvectors with the optimized weighing factors: C1 = 0.5532, C2 = 1.0000, C3 = 0.4106 and,
Fig. 15. Natural frequencies of the optimum shaped beam as a function of the maximum relative deformation. (a) evolution of natural frequencies; (b) ratio of the objective function and initial objective function.
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 16. Beam shaped with the optimum geometry for maximum deformation of 4 mm. (a) optimum shape; (b) shaped aluminum beam.
Table 8 Experimental natural frequencies of the aluminum beam with the optimum shape for maximum deflection of 4 mm. Comparison to the expected theoretical values.
1 2 3 3
Experimental (Hz)
Numerical (Hz)
Error (%)
50.0 7 0.5 153.5 70.5 252.0 7 0.5 346.0 7 1.0
51.57 153.15 252.91 351.15
3.0 þ0.2 0.4 1.5
C4 = 0.3033. The obtained shape of the beam was measured with a coordinate measuring machine and the results are shown in Fig. 16(a) whose measurement errors are70.1 mm. By testing experimentally the beam shaped with the optimum geometry, following the same experimental procedure described in Section 5, we obtained the results presented in Table 8. As we can see, the error between the expected theoretical values and the real experimental values remained below 3%. Again, the effectiveness of the method to find the optimized shape of the structure is clear. In this case, there was no preponderance of a given eigenvector in the optimized shape of the beam. Therefore, the optimization procedure was necessary (there was no viable educated guess).
6. Conclusions The optimization of natural frequencies by shaping the structure with a geometry obtained from a linear combination of mode shapes is feasible. The method is effective in maximizing and/or placing the natural frequencies at desired values. The parametric analysis of the method shows that:
the adopted boundary condition plays an important role. The beam must be axially constrained to achieve significant alteration of the natural frequencies;
by shaping the beam with the geometry of a given mode shape, we increase the natural frequency associated to this mode shape, predominantly, and frequencies below this natural frequency remain unaffected. Hence, the maximization of a given natural frequency can be done by shaping the beam with the geometry of the mode shape associated to that frequency. This represents an educated guess and no optimization procedure is necessary because the obtained results are very close to those obtained with the optimization algorithm. This can be useful to increase the stiffness of the structure, by shifting the first natural frequency towards higher frequencies. Also, this can be useful to create dead zones in the structure, by shifting a given natural frequency towards higher frequencies and, increasing the ”gap” in the spectrum between two natural frequencies. On the other hand, the placement of several natural frequencies at desired values requires the use of the optimization algorithm.
Acknowledgement This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico [grant number 304709/ 2015-8]. Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes Guilherme Augusto Lopes da Silva, Rodrigo Nicoletti n University of São Paulo, São Carlos School of Engineering, Dept. of Mechanical Engineering, Trabalhador São-Carlense 400, 13566-590 São Carlos, Brazil
a r t i c l e i n f o
abstract
Article history: Received 1 June 2016 Received in revised form 10 February 2017 Accepted 24 February 2017 Handling Editor: M.P. Cartmell
This work focuses on the placement of natural frequencies of beams to desired frequency regions. More specifically, we investigate the effects of combining mode shapes to shape a beam to change its natural frequencies, both numerically and experimentally. First, we present a parametric analysis of a shaped beam and we analyze the resultant effects for different boundary conditions and mode shapes. Second, we present an optimization procedure to find the optimum shape of the beam for desired natural frequencies. In this case, we adopt the Nelder-Mead simplex search method, which allows a broad search of the optimum shape in the solution domain. Finally, the obtained results are verified experimentally for a clamped-clamped beam in three different optimization runs. Results show that the method is effective in placing natural frequencies at desired values (experimental results lie within a 10% error to the expected theoretical ones). However, the beam must be axially constrained to have the natural frequencies changed. & 2017 Published by Elsevier Ltd.
Keywords: Natural frequency Optimization Embossed pattern Finite element method Mechanical vibration Slender beams
1. Introduction Manufactured products can present excessive vibration during operation due to the inherent dynamics of their structure and due to the source of excitation. Structure dynamics affects vibration depending on the location of resonance frequencies in the frequency range of operation, on the damping level of the structure, on the appearance of non-linear modes during operation, and on the presence of vibro-acoustic coupling. The source of excitation affects vibration depending on the amplitude level of the load, which can lead the system to high vibration levels even in low frequency response regions of the structure. Unless the Engineer has control of the excitation source, the problem of reducing the vibratory response of a system lies on the modification of the structure, i.e. on the modification of its dynamics. The dynamics of a structure can be modified either actively or passively. An active modification of structure dynamics implies the adoption of sensors and actuators in a control loop feedback. In this case, the fine-tuning of the controller's gains moves the poles and zeros of the structure to new positions in the complex plane, thus altering the structure dynamic characteristics in closed loop. The passive modification refers to a physical change of the structure in terms of its geometry and/or its material, thus affecting the inherent mass and stiffness of the system and, consequently, affecting its dynamic characteristics. In both cases, we can shift natural frequencies, placing them at desired values to avoid operating frequency regions or to create frequency dead zones in the structure. n
Corresponding author. E-mail addresses: [email protected] (G.A.L.d. Silva), [email protected] (R. Nicoletti).
http://dx.doi.org/10.1016/j.jsv.2017.02.053 0022-460X/& 2017 Published by Elsevier Ltd.
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Nomenclature Ci E desired fi k fi nd fi F h K L M
i-th weighing factor (–) material Young modulus (N.m −2) i-th desired natural frequency (Hz) i-th natural frequency at k-th iteration (Hz) i-th natural frequency of non-deformed structure (Hz) objective function (–) height of cross section (m) stiffness matrix beam length (m) inertia matrix
T ui ^ u i u, v w x
α δmax ρ θ φ ^ φ
transformation matrix i-th eigenvector (–) i-th normalized eigenvector (–) nodal displacements (m) width of cross section (m) vector of degrees of freedom beam finite element tilting angle (rad) maximum deformation of the beam (m) material density (kg.m −3) nodal angular displacement (rad) shape of the beam (–) normalized shape of the beam (–)
In this work, we will focus on the placement of natural frequencies of the structure to desired frequency regions. More specifically, we will focus on the placement of natural frequencies of a beam by changing its shape (passive modification of the structure). The optimization of beams subjected to frequency restriction has been studied since the late 1960s [1–3]. However, such studies focused on parameter optimization (not shape optimization). The actual idea of optimizing the shape of beams and plates aiming at desired eigenfrequencies started in the 1990s by changing the local thickness of the beam or plate [4,5]. For that, the authors defined the thicknesses of different regions of the structure as design variables of the optimization procedure. Finite element modeling was naturally suitable for the optimization procedure because this numerical method divides the structure in discrete regions of constant thickness. Experimental results showed the effectiveness of the methodology in placing the natural frequencies of the structure at desired specific values [6,7]. The drawback of changing the thickness of the structure lies on practical difficulties of manufacturing. In this case, machining is a cumbersome process and stamping (a more feasible solution) requires a careful design of the dies. Alternatively, we can change the geometry of the beam or plate considering constant thickness. For example, the coordinates of points in the boundary of the structure can be used as design variables in the optimization procedure [8]. Hence, the geometry changes in the plane defined by the larger dimensions of the beam or plate (thickness is perpendicular to this plane and it is kept constant). Again, we can shift natural frequencies of the structure towards higher values, but this procedure cannot be applied to structures with fixed boundaries (e.g. welded or bolted plates at their whole perimeter). If the boundaries are fixed but the plate has holes whose geometry can be changed, we can optimize the shape of the holes instead, as numerically shown in [9]. In this case, one reached a 20% increase of the first natural frequency and a 12% increase of the fourth natural frequency of the plate in two different optimization runs. We can also design optimal reinforcements to the structure with constant thickness, where the structure geometry remains the same. In [10], the adopted design variables were the length, orientation, and coordinates of the reinforcement to be applied perpendicular to the plane of a plate. In this case, the optimization procedure was applied to find the best position and length of the reinforcement that maximizes the first natural frequency of the plate. The obtained results showed an increase of 10% of the first natural frequency of the plate. Other examples of optimal positioning of reinforcements towards structure dynamic modification are found in [11,12]. These works deal with reinforcements with constant height, but it is also possible to optimize the shape of the reinforcements. In [13], the position and length of the reinforcements are known and fixed, and the shape of the reinforcements (height) is optimized towards maximization of the first natural frequency of the plate. The results showed an increase of near 40% of the first natural frequency of the plate in study. Another way of changing the inherent dynamics of beams and plates with constant thickness is through embossing patterns on them. In this case, the structure is locally deformed to increase stiffness and, consequently, the natural frequencies. In [14], an optimization procedure finds the best locations of grooves in the surface of a plate. The results show an improve of ∼270% of the first natural frequency, and the final deformed shape of the plate (optimum shape) is very suitable for sheet metal stamping. In [15] and [16], the grooves are defined by the mode shapes of the structure in original condition (not deformed), and the final deformed shape of the structure is a linear combination of these mode shapes. The authors used a three-level full factorial design of experiments (DOE) to find the weighing factors of the linear combination of mode shapes that shifted the natural frequencies towards the desired values. Such DOE approach is suitable for time consuming computational models, with high number of degrees-of-freedom. The present work gives an additional contribution to the embossed pattern methodology presented in [15] and [16]. Here, we further investigate the effects of combining mode shapes to shape a beam to change its natural frequencies, both numerically and experimentally. First, we analyze the resultant effects for different boundary conditions and mode shapes. This analysis allows a better insight into the phenomenology of the problem. Second, we present an optimization procedure to find the optimum shape of the beam for desired natural frequencies that is different from the one used in [16]. In this case, we adopt the Nelder-Mead simplex search method, which allows a much broader search of the optimum shape in the Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 1. Degrees-of-freedom of the finite element used to model the beam.
solution domain than the design of experiments adopted by [16], for models whose tests are not time consuming (cheap and quick to calculate). Finally, the obtained results are verified experimentally for a clamped-clamped beam. We performed three different optimization runs and the results show that the experimental natural frequencies of the shaped beam lie within a 10% error to the expected theoretical ones.
2. Mathematical modeling of the shaped beam This work focuses on the natural frequencies of permanently deformed beams (shaped beams) and, on the optimum deformation (shape) of the beam that results in desired natural frequency values. Hence, the beams in analysis are not straight and they have a certain known geometry. We will neglect in the mathematical modeling any residual stress in the beam due to the plastic deformation process (it is assumed that the beam went through a stress relief process). The mathematical modeling of the beam is based on Euler-Bernoulli finite elements with axial nodal displacements [17]. The axial nodal displacements are important to account for the coupling effects that arise when the elements of the model present a relative angle between each other. Hence, the vector of degrees-of-freedom of the finite element in local coordinates is:
x e = { u1 v1 θ1 u2
v2
θ2}
(1)
where ui is the in-plane (axial) displacement of the i-th node of the element, vi is the out-of-plane (perpendicular) displacement of the i-th node of the element, and θi is the angular displacement of the i-th node of the element (Fig. 1). In the case of a non-straight beam (shaped beam), the beam is modeled by straight elements with a relative angle between each other to follow the shape of the beam. Hence, the elements of the model will present a tilting angle α in relation to the global coordinate system (Fig. 2). Hence, we must perform a coordinate transformation to obtain the global matrices of the model in global coordinates (X,Y):
MGe = T T MeT
(2)
K Ge = T T K eT
(3)
MGe
K Ge
where and are the global inertia and stiffness matrices of the element, Me and K e are the inertia and stiffness matrices of the element in local coordinates, and T is the transformation matrix (all these matrices are found in [17]). By arranging the global matrices of the elements, we derive the equations of motion of the beam:
(4)
Mx¨ + Kx = 0
where M is the inertia matrix of the beam model, K is the stiffness matrix of the beam model, x¨ is the acceleration vector, and x is the displacement vector. We will not include damping in the model for simplicity. This assumption is acceptable for slender beams, which usually present low damping factors (below 5%). It is also important to emphasize that the adopted mathematical modeling, based
Fig. 2. Example of shaped beam: (a) original geometry, (b) finite element model.
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Table 1 Boundary conditions of the beam adopted in the analysis. Free-free (FF)
Cantilever (CB)
Simply supported axial free (SSF)
Simply supported (SS)
Clamped-clamped (CC)
Table 2 Properties of the baseline beam in the analyses. Length (L)
0.600
m
Width (w) Height (h) Density (ρ) Young modulus (E)
0.030 0.003 2,688 69 109
m m kg.m 3 N.m 2
on Euler-Bernoulli finite elements, are only valid for slender beams. Thus, considering Eq. (4), we can solve the associated eigenvalue problem and find the eigenvalues (λi) and eigenvectors (u i ) of the system. The adopted boundary conditions are listed in Table 1 and the properties of the baseline beam are listed in Table 2. The baseline beam has a rectangular cross section and it has a height-to-width ratio (h/w ) of 0.1 and a height-to-length ratio (h/L ) of 0.005. The number of finite elements in the model of the beam affects the resultant eigenvalues (natural frequencies) and, a convergence analysis was performed. In this case, there was no significant variation of the first five natural frequencies of the beam model when adopting 36 finite elements or more, irrespective of the adopted boundary condition. Hence, we will adopt 36 finite elements in the model in the following analyzes.
3. Parametric analysis of the deformed beam In this section, we will change the shape of the baseline beam and analyze the effects on the resultant natural frequencies, using the mathematical model described before. For that, we will use the eigenvectors (mode shapes) of the baseline beam (straight beam) as shape functions. That means, the beam will take the geometry of a given mode shape of the baseline beam, depending on the adopted boundary condition. Considering that an eigenvector only gives relative displacements, we must define the scale with which we will shape the beam. This scale is defined by the maximum deformation of the shaped beam (δmax in Fig. 3). In this case, the beam is
Fig. 3. Maximum deflection of the shaped beam in relation to the baseline beam.
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shaped following the chosen eigenvector in a proportion that the maximum distance between the shaped beam and the baseline beam is δmax. Thus, by changing the value of δmax, the geometry of the whole beam is changed proportionally to the chosen eigenvector (mode shape). The maximum deformation of the shaped beam (δmax) gives the information of how much deformed is the beam. In the following analyses, we will adopt a different parameter to inform the level of deformation of the shaped beam: the maximum relative deformation. The maximum relative deformation is the ratio between δmax and the height of the beam cross section (h). Hence, if the maximum relative deformation is 1.0, it means that the maximum distance between the shaped beam and the baseline beam is one height of the beam cross section. 3.1. Effect of boundary condition Here, the baseline beam will be shaped by its first mode shape, depending on the adopted boundary condition. The maximum relative deformation of the beam varied from 0 (straight beam) to 3.0 (maximum distance between the shaped beam and the baseline beam is three heights of the beam cross section). The results are presented in Fig. 4. As we can see, in the cases of free-free, cantilever and, simply supported axial free boundary conditions (Fig. 4(a) to (c)), there is no significant variation of the first four natural frequencies of the shaped beam in relation to those of the baseline beam (straight beam). In the case of the simply supported boundary condition (Fig. 4(d)), the value of the first natural frequency of the beam significantly increases as the maximum relative deformation increases. For maximum relative deformations above 1.5, there is a shift in the order of the mode shapes (the first mode shape becomes the second mode shape, and vice-versa). In the case of the clamped-clamped boundary condition (Fig. 4(e)), not only the first natural frequency increases as the maximum relative deformation increases, but also the third and fifth natural frequencies present some variation towards higher values. The order of the modes changes for maximum relative deformations above 2.5. The results in Fig. 4 show that the shaped beam must be axially constrained to present variations in its natural frequencies. In addition, by shaping the beam in its first mode shape geometry, we affect the first natural frequency predominantly. In the case of the clamped-clamped beam, we also see effects on the third and fifth natural frequencies, whose mode shapes are from the same family of the first mode shape (even functions). 3.2. Effect of mode shape geometry Considering that, the first natural frequency is predominantly affected by shaping the beam in its first mode shape, we adopt other mode shapes to verify their effect on the natural frequencies. Figs. 5(a) and 5(b) present the first five natural frequencies of the baseline beam shaped by its second mode shape geometry, depending on the adopted boundary condition. The shaped beams not axially constrained (free-free, cantilever, simply supported axial free) presented again no variation in their natural frequencies and the results are omitted. In the case of the simply supported boundary condition (Fig. 5(a)), the value of the second natural frequency of the beam significantly increases as the maximum relative deformation increases. The order of the mode shapes also changes as the value of the natural frequency associated to the second mode shape surpasses the succeeding natural frequencies. The value of the first natural frequency is not affected, as well as the values of all other natural frequencies analyzed. In the case of the clamped-clamped boundary condition (Fig. 5(b)), not only the second natural frequency increases as the maximum relative deformation increases, but also the fourth natural frequency present some variation towards higher values. The order of the modes changes as the maximum relative deformation increases. The value of the first natural frequency is also not affected, as well as the values of all other odd natural frequencies analyzed (third and fifth). Figs. 5(c) and 5(d) present the first six natural frequencies of the baseline beam shaped by its third mode shape geometry, depending on the adopted boundary condition. The shaped beams not axially constrained (free-free, cantilever, simply supported axial free) presented again no variation in their natural frequencies and the results are omitted. In the case of the simply supported boundary condition (Fig. 5(c)), the value of the third natural frequency of the beam significantly increases as the maximum relative deformation increases. Again, the order of the mode shapes changes as the value of the natural frequency associated to the third mode shape surpasses the succeeding natural frequencies. The values of the first and second natural frequencies are not affected, as well as the values of all other natural frequencies analyzed. In the case of the clamped-clamped boundary condition (Fig. 5(d)), not only the third natural frequency increases as the maximum relative deformation increases, but also the fifth natural frequency present some variation towards higher values. The order of the modes changes as the maximum relative deformation increases. The values of the first and second natural frequencies are not affected, as well as the values of all other even natural frequencies analyzed (fourth and sixth). The results in Fig. 5 show that by shaping the beam in its i-th mode shape geometry, we predominantly affect the i-th natural frequency. The natural frequencies of the baseline beam whose values are below this i-th natural frequency remain unaffected. In the case of the clamped-clamped boundary condition, we also see the effects on the natural frequencies whose mode shapes are from the same family of the i-th mode shape. In this case, frequency veering is observed when the frequency of the i-th mode reaches the frequency of a mode of the same family. Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 4. Natural frequencies of the shaped beam (first mode shape geometry) as a function of the maximum relative deformation: (a) free-free, (b) cantilever, (c) simply supported axial free, (d) simply supported, (e) clamped-clamped.
4. Optimization procedure The results presented in the previous section show that it is possible to increase a natural frequency of the beam by shaping it with the geometry of a mode shape of the beam in its original condition (straight beam). The question that follows is whether two or more mode shapes can be combined to change more natural frequencies of the beam simultaneously. The answer is yes, as demonstrated in [15,16]. Actually, the problem is how to combine the mode shapes to find the Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 5. Natural frequencies of the shaped beam as a function of the maximum relative deformation: (a) simply supported (2nd. mode geometry), (b) clamped-clamped (2nd. mode geometry), (c) simply supported (3rd. mode geometry), (d) clamped-clamped (3rd. mode geometry).
geometry of the beam that results in desired natural frequencies. In [16], the authors did a linear combination of the mode shapes, whose coefficients (weighting factors of the linear combination) were obtained by Design of Experiments, an exhaustive try out process. Here, we present an optimization procedure to find these coefficients of the linear combination of mode shapes. Consider the mathematical model of the beam (Eq. (4)). We can solve the associated eigenvalue problem and find the eigenvalues (λi) and eigenvectors (u i ) of the system. Considering that the eigenvectors are linearly independent vectors, we can use them as a basis to find the optimized shape of the beam, in the form: N
φ=
∑ Ciui i=1
(5)
where C i is the weighing factor of the i-th mode shape of the basis, N is the number of mode shapes that form the basis, and φ is the final geometry of the shaped beam. The task of the optimization procedure is to find appropriated values for the weighing factors C i that make the beam, shaped in the geometry φ, present the desired natural frequencies. The problem with Eq. (5) is that eigenvectors represent mathematically a direction in space [18]. Hence, the elements of the eigenvectors have no absolute values, and we must adopt a norm to represent them. In the present case, the elements of the eigenvectors are normalized by the respective maximum absolute value:
^ = u i
ui max (|ui |)
(6)
^ is the i-th normalized eigenvector of the system. Thus, the normalized eigenvectors will always present element where u i values between -1 and 1. Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Another problem with Eq. (5) is that the function φ can suffer from scaling and, consequently, it can grow indefinitely for the same linear combination of eigenvectors. If the weighing factors Ci are scaled by a scalar a, the resultant shape will be:
φ* = aφ
(7)
where φ* is a scaled version of the shape φ. If a tends to infinity, the deformed shape will also tend to infinity. However, both deformed shapes represent the same linear combination of eigenvectors (same family of shapes). To avoid such infinite number of similar solutions to the problem, we also normalize the shape by the respective maximum absolute value: N
φ=
∑ Ciu^ i
(8)
i=1
^=δ φ max
φ max (|φ|)
(9)
where δmax is the maximum deformation of the beam (Fig. 3). Thus, the normalized shape will present element values between −δmax and δmax. The maximum deformation of the beam is defined previously to the optimization procedure (parameter of design). ^ , we can find the respective vector of tilting angles of the finite element model After calculating the normalized shape φ (vector α ). The finite element model of the beam is then updated and the associated eigenvalue problem will result in new eigenvalues and eigenvectors of the system. In this case, the new eigenvalues (natural frequencies) will refer to the shaped ^. beam in the geometry given by φ Hence, we can write an optimization algorithm whose flowchart is shown in Fig. 6. Initially, we solve the eigenvalue problem of the system in original condition (straight beam) and we obtain the eigenvectors of the system. These eigenvectors are normalized according to Eq. (6) and they form the basis for the shape optimization. Given an initial set of weighing factors, the looping of optimization starts. We calculate the new geometry of the beam with Eqs. (8) and (9). The finite element model of the beam is updated with the new geometry and, the eigenvalue problem is solved to find the natural frequencies of the shaped beam. We use the natural frequencies of the shaped beam with the new geometry to calculate the objective function F.
Fig. 6. Flowchart of the optimization algorithm.
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In this work, we adopt three different objective functions to perform the shape optimization of the beam. If the objective is the maximization of the first natural frequency of the beam, then we must minimize the objective function:
f 1nd
F (k ) =
f1k
(10)
k f1
where is the first natural frequency of the shaped beam calculated in the k-th iteration of the algorithm, and f 1nd is the first natural frequency of the baseline beam (straight beam). If the objective is the maximization of the j-th natural frequency of the structure keeping the other natural frequencies unchanged, then we must minimize the objective function: j−1 ⎛
F (k ) =
∑ ⎜⎜ 1 − i=1
⎝
2
f nd f ind ⎞ ⎟ + j + ⎟ fik ⎠ f kj
2 ⎛ f nd ⎞ ⎜1 − i ⎟ ⎜ ⎟ fik ⎠ i=j+1 ⎝ N
∑
(11)
k fi
nd fi
where is the i-th natural frequency of the shaped beam calculated in the k-th iteration of the algorithm, and is the i-th natural frequency of the baseline beam (straight beam). If the objective is the positioning of natural frequencies at specific desired values, then we must minimize the objective function:
F (k ) =
N
⎛
i=1
⎝
∑ ⎜⎜ 1 −
2
f idesired ⎞ ⎟ ⎟ fik ⎠
(12)
desired fi
where is the desired value for the i-th natural frequency of the shaped beam. After calculating the value of the objective function (Fig. 6), we implement the algorithm fminsearch of software MATLAB to find new values of the weighing factors. The algorithm fminsearch performs an unconstrained nonlinear optimization (minimization of the objective function) using the Nelder-Mead simplex search method [19]. This is a direct search method that does not use numerical or analytical gradients. In addition, it may only give local optimum solutions. The adopted convergence criteria are the tolerance value of the objective function (10 7 in the present work), and the tolerance value of the design variables, i.e. the tolerance value of the weighing factors (10 5 in the present work). The adopted stop criterion is the maximum number of iterations (300 iterations in the present work).
5. Experimental results We performed the optimization analysis in an aluminum beam whose properties are listed in Table 3. Initially, we determined experimentally the natural frequencies and mode shapes of the beam under a clamped-clamped boundary condition. For that, the beam (Fig. 7–1) was mounted on fixed clamping blocks (Fig. 7–2), and a electrodynamic shaker (Fig. 7–3) was connected to the beam 100 mm apart from one of the clamped ends. A load cell mounted in the shaker side of the stinger (Fig. 7–4) measured the excitation force. The vibrating response of the beam under excitation was measured by a laser vibrometer (Fig. 7–5) at every 50 mm of the beam (11 equally spaced measuring points). The excitation signal used in the experiment was a chirp signal, ranging from 10 to 400 Hz and back to 10 Hz in 8 s (linear variation). This signal was repeated 10 times, thus resulting in a total acquisition period of 80 s. The adopted acquisition rate was 5 kHz. The frequency response functions were calculated by the H1 and H2 estimators [20], with Hanning window of 5,000 points and overlap of 3,500 points. Table 4 presents the obtained natural frequencies and associated mode shapes of the beam in the frequency range of study (below 400 Hz). The measurement error of 0.5 Hz represents the measurement accuracy of the system. We built the mathematical model of the beam using the Euler-Bernoulli beam finite elements described in Section 2. We adopted 36 finite elements to represent the beam under clamped-clamped boundary condition. The beam was in its original condition (straight beam) and, consequently, all finite elements had null tilting angle (α ¼ 0). The parameter values used in the model were the same as those presented in Table 3. By mounting the global inertia and stiffness matrices and solving the Table 3 Properties of the aluminum beam in analysis. Property
Value
Unit
Total length Width (b) Thickness (h) Material density
0.6 31 3 2688
m mm mm
Young modulus (E)
kg.m 9
69 ×10
N.m
−3
−2
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 7. Test rig of the clamped-clamped beam: (1) beam, (2) clamping blocks, (3) shaker, (4) load cell, (5) laser vibrometer.
Table 4 Experimental natural frequencies of the aluminum beam under clamped-clamped boundary condition.
1 2 3 3
Natural frequency (Hz)
Mode shape
42.57 0.5 107.5 7 0.5 195.5 7 0.5 340.57 0.5
First bending Second bending Third bending Fourth bending
Table 5 Comparison between experimental and numerical natural frequencies of the beam (straight beam).
1 2 3 3
Numerical (Hz)
Experimental (Hz)
Error (%)
42.14 105.22 195.73 339.68
42.5 70.5 107.5 7 0.5 195.5 7 0.5 340.5 70.5
0.8 2.1 þ 0.1 0.2
eigenvalue problem of the system, we obtained the numerical results shown in Table 5. These results were obtained after including an additional lumped mass of 40 g in the degrees-of-freedom of the model related to the position of the attached shaker (node located 100 mm apart from the clamped edge). Such addition of a lumped mass in the model accounted for the stinger attached to the structure. We observe that the numerical results of natural frequencies present good correlation to the experimental ones. Such good correlation can also be noted in the resulting mode shapes (Fig. 8). These mode shapes presented in Fig. 8 form the basis of eigenvectors in the optimization procedure applied to the present case (clamped-clamped beam). Considering that the mathematical model is correlated to the real structure, it was used in the optimization algorithm. Here, we performed three different optimization analyzes:
maximization of the first natural frequency, where the objective was to increase the global stiffness of the structure; maximization of the third natural frequency, keeping the first and the second natural frequencies unchanged, where the objective was to create a gap of resonances in the structure (dead zone);
positioning of natural frequencies, where the objective was placing the four natural frequencies at desired values. In all these analyzes, the maximum deformation of the shaped beam was a parameter previously defined. Therefore, the results are presented as a function of the maximum deformation of the beam (δmax). In addition, the initial guess of the weighing factors was null in all cases (Ci(0) = 0, ∀ i ). All four eigenvectors were used in the shape optimization (N ¼4), irrespective of the number of natural frequencies involved in the objective function. 5.1. Maximization of the first natural frequency The optimization procedure to maximize the first natural frequency of the beam adopted the objective function of Eq. (10), where f 1nd = 42.14 Hz (first natural frequency of the straight beam). The obtained numerical results are presented in Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 8. Comparison between the experimental and numerical mode shapes of the beam (straight beam): (a) first mode, (b) second mode, (c) third mode, (d) fourth mode.
Fig. 9. Natural frequencies of the optimum shaped beam as a function of the maximum relative deformation. (a) evolution of natural frequencies; (b) ratio of the objective function and initial objective function.
Fig. 9. As we can see, the optimum values depend on the maximum relative deformation of the beam. The first natural frequency increases as the maximum relative deformation increases, thus showing a similar behavior to what has been shown in the parametric analysis (Section 3). However, when the first natural frequency reaches the value of the second natural frequency, there is no significant change in the results, irrespective of the maximum relative deformation adopted. In this case, the first natural frequency of the beam remains the same for any maximum relative deformation above 2, and the value of the objective function remains constant (Fig. 9(b)). Considering these results, we manufactured an aluminum beam shaped with the optimum geometry with maximum deformation of 10 mm (max. relative deformation of 3.3). The optimum geometry is shown in Fig. 10, which is the result of a linear combination of the eigenvectors with the optimized weighing factors: C1 = 1.0000, C2 = 0.0346, C3 = 0.0983 and, C4 = 0.1292. The obtained shape of the beam was measured with a coordinate measuring machine and the results are shown in Fig. 10(a) whose measurement errors are70.1 mm. Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 10. Beam shaped with the optimum geometry for maximum deformation of 10 mm. (a) optimum shape; (b) shaped aluminum beam.
Table 6 Experimental natural frequencies of the beam shaped with the optimum geometry with maximum deformation of 10 mm. Comparison to the expected theoretical values.
1 2 3 3
Experimental (Hz)
Numerical (Hz)
Error (%)
98.5 7 0.5 107.5 7 0.5 265.0 7 1.0 368.57 0.5
105.23 105.24 261.62 403.97
-6.4 þ 2.3 þ 1.3 -8.8
By testing experimentally the beam shaped with the optimum geometry, following the same experimental procedure described in Section 5, we obtained the results presented in Table 6. As we can see, the error between the expected theoretical values and the real experimental values remained below 9%. This shows the effectiveness of the method to find the optimized shape of the structure that maximizes its first natural frequency. Considering that the mass of the structure remains unchanged, the stiffness of the structure is maximized. Looking at the resulting optimum geometry (Fig. 10) and at the optimized weighing factors, we can see that there is a preponderance of the first mode shape in the solution. Using the first eigenvector only as the geometry of the beam, we obtained the results shown in Fig. 11. As we can see, the results are very similar to those obtained in the optimization procedure using the complete basis of four eigenvectors (Fig. 9). This shows that, shaping the beam with the geometry of its first mode shape is as efficient as using a linear combination of the first four mode shapes in the maximization of the first natural frequency of the structure. In fact, as already demonstrated in Section 3, when the structure has the geometry of a given mode shape, we maximize the associated natural frequency of this mode shape. Hence, to maximize the first natural frequency, we can shape the beam with the geometry of the first mode shape of a straight beam. This represents an educated guess without performing the optimization analysis. However, it is important to emphasize that, there will be no control of the remainder natural frequencies, and they can change freely towards higher values.
Fig. 11. Natural frequencies of the shaped beam as a function of the maximum relative deformation (geometry of the first eigenvector).
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5.2. Maximization of the third natural frequency The optimization procedure to maximize the third natural frequency of the beam, keeping the first and the second natural frequencies unchanged, adopted the objective function of Eq. (11), where f 1nd = 42.14 Hz , f 2nd = 105.22, and f 3nd = 195.73 (first, second, and third natural frequencies of the correlated model). The obtained numerical results are presented in Fig. 12. Again, the optimum results depend on the maximum relative deformation of the beam. The third natural frequency value increases as the maximum relative deformation increases, in a similar way to what has been shown in Section 3, where only one eigenvector is used as the geometry of the shaped beam (third eigenvector). In this case, the third natural frequency of the beam remains the same for any maximum relative deformation of the optimum shape above 1. Consequently, the value of the objective function remains constant above this threshold value (Fig. 12(b)). Considering these results, we manufactured an aluminum beam shaped with the optimum geometry for a maximum deformation of 4 mm (max. relative deformation of 1.3). The optimum geometry is shown in Fig. 13, which is the result of a linear combination of the eigenvectors with the optimized weighing factors: C1 = 0.9140, C2 = 0.2906, C3 = 1.0000 and, C4 = 0.0888. The obtained shape of the beam was measured with a coordinate measuring machine and the results are shown in Fig. 13(a) whose measurement errors are70.1 mm.
Fig. 12. Natural frequencies of the optimum shaped beam as a function of the maximum relative deformation. (a) evolution of natural frequencies; (b) ratio of the objective function and initial objective function.
Fig. 13. Beam shaped with the optimum geometry for maximum deformation of 4 mm. (a) optimum shape; (b) shaped aluminum beam.
Table 7 Experimental natural frequencies of the beam shaped with the optimum geometry for maximum deformation of 4 mm. Comparison to the expected theoretical values.
1 2 3 3
Experimental (Hz)
Numerical (Hz)
Error (%)
43.5 7 0.5 106.57 0.5 308.5 7 0.5 347.8 7 0.5
42.17 105.21 339.70 339.71
þ3.2 þ1.2 -9.2 þ2.3
Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Fig. 14. Natural frequencies of the shaped beam as a function of the maximum relative deformation (geometry of the third eigenvector).
By testing experimentally the beam shaped with the optimum geometry, following the same experimental procedure described in Section 5, we obtained the results presented in Table 7. As we can see, the error between the expected theoretical values and the real experimental values remained below 10%. Looking at the optimum geometry (Fig. 13) and at the optimum weighing factors, we can see that there is a preponderance of the first and third mode shapes in the solution. However, the results of natural frequencies were very similar to those using the third mode only (Fig. 14). Again, the third mode shape of the straight beam represents an educated guess to maximize the third natural frequency of the structure, keeping the first and second natural frequencies unchanged (no need of an optimization analysis). In both cases (shape geometry composed of a full set of eigenvectors or of the third eigenvector only), we obtained a frequency range of ∼200 Hz without any resonances in the beam (from ∼106 Hz to ∼310 Hz), thus showing the effectiveness of the method. 5.3. Positioning of natural frequencies The optimization procedure for positioning the natural frequencies of the beam at desired values adopted the objective function of Eq. (12). The desired values are 50, 150, 250, and 350 Hz, which means that all natural frequencies should be changed. Fig. 15 presents the optimum results as a function of the maximum relative deformation of the beam. As we can see, the optimization algorithm achieves the objective for maximum relative deformations between 1 and 2.3 (Fig. 15(b)). Above the value of 2.7, there are some oscillations in the solution, thus indicating that there are regions of optimality depending on the maximum relative deformation of the beam. Considering these results, we manufactured an aluminum beam shaped with the optimum geometry for a maximum deformation of 4 mm (max. relative deformation of 1.3). The optimum geometry is shown in Fig. 16, which is the result of a linear combination of the eigenvectors with the optimized weighing factors: C1 = 0.5532, C2 = 1.0000, C3 = 0.4106 and,
Fig. 15. Natural frequencies of the optimum shaped beam as a function of the maximum relative deformation. (a) evolution of natural frequencies; (b) ratio of the objective function and initial objective function.
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Fig. 16. Beam shaped with the optimum geometry for maximum deformation of 4 mm. (a) optimum shape; (b) shaped aluminum beam.
Table 8 Experimental natural frequencies of the aluminum beam with the optimum shape for maximum deflection of 4 mm. Comparison to the expected theoretical values.
1 2 3 3
Experimental (Hz)
Numerical (Hz)
Error (%)
50.0 7 0.5 153.5 70.5 252.0 7 0.5 346.0 7 1.0
51.57 153.15 252.91 351.15
3.0 þ0.2 0.4 1.5
C4 = 0.3033. The obtained shape of the beam was measured with a coordinate measuring machine and the results are shown in Fig. 16(a) whose measurement errors are70.1 mm. By testing experimentally the beam shaped with the optimum geometry, following the same experimental procedure described in Section 5, we obtained the results presented in Table 8. As we can see, the error between the expected theoretical values and the real experimental values remained below 3%. Again, the effectiveness of the method to find the optimized shape of the structure is clear. In this case, there was no preponderance of a given eigenvector in the optimized shape of the beam. Therefore, the optimization procedure was necessary (there was no viable educated guess).
6. Conclusions The optimization of natural frequencies by shaping the structure with a geometry obtained from a linear combination of mode shapes is feasible. The method is effective in maximizing and/or placing the natural frequencies at desired values. The parametric analysis of the method shows that:
the adopted boundary condition plays an important role. The beam must be axially constrained to achieve significant alteration of the natural frequencies;
by shaping the beam with the geometry of a given mode shape, we increase the natural frequency associated to this mode shape, predominantly, and frequencies below this natural frequency remain unaffected. Hence, the maximization of a given natural frequency can be done by shaping the beam with the geometry of the mode shape associated to that frequency. This represents an educated guess and no optimization procedure is necessary because the obtained results are very close to those obtained with the optimization algorithm. This can be useful to increase the stiffness of the structure, by shifting the first natural frequency towards higher frequencies. Also, this can be useful to create dead zones in the structure, by shifting a given natural frequency towards higher frequencies and, increasing the ”gap” in the spectrum between two natural frequencies. On the other hand, the placement of several natural frequencies at desired values requires the use of the optimization algorithm.
Acknowledgement This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico [grant number 304709/ 2015-8]. Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i
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Please cite this article as: G.A.L.d. Silva, R. Nicoletti, Optimization of natural frequencies of a slender beam shaped in a linear combination of its mode shapes, Journal of Sound and Vibration (2017), http://dx.doi.org/10.1016/j.jsv.2017.02.053i