Dynamic vibration absorbers (DV As) and tuned mass-dampers (TV As) have wide-spread applications in the aerospace industry, the automotive sector, and in civil engineering structures. There are numerous designs of active and passive vibration attenuators or absorbers that isolate structural vibrations at or around the desired frequency. All these design approaches are fundamentally di erent ways to modify and tune the placement of the resonant frequencies of the host structure. The current work presents a novel method to passively attenuate vibration over a broad frequency bandwidth in the presence of uncertainty. An array of linear oscillators, also referred to as subordinate oscillator arrays (SOAs), are attached to a two-degrees-of-freedom structure to produce an attenuated broadband frequency response around a target frequency. SOAs can also be interpreted as an array of DV As and in some categories, they can be considered as an approach to meta-structures. Another objective of the current work is to develop a hands-on approach to extend classroom teaching of vibration-isolation using SOAs made out of fettuccine strands and modeling clay. The frequencies of the oscillators in the array are tuned by varying the length of each strand and the mass of the modeling clay attached to its tip. Uncertainty in dynamic properties of such oscillators often results in mistuned SOAs with non-uniform frequency response function. Therefore, designing and testing fettuccine-based SOAs allows students to handle cases when structural uncertainties arise in engineering systems. Additionally, some of the work in the ﬁeld of meta-structures can be modeled and represented by SOAs and this will provide a straight forward way to teach students some of these contemporary concepts. Keywords: Vibration Isolation, Subordinate Oscillator Array, Fettuccine Pasta, Modeling Clay, Structural Dynamics, Education, Dynamic Vibration Absorbers 1. Introduction For more than a century, dynamic vibration absorbers (DV As) have attenuated vibratory response near the natural frequency of engineering structures [1]. The fundamental idea behind DV As is to im- prove the dynamics close to the resonance frequency of the host structure by securing a secondary dy- namic system (a subordinate system) tuned to this frequency. Di erent applications have embraced and Corresponding author Email address: saitejp@vt.edu (Sai Tej Paruchuri) 1These authors have contributed equally Preprint submitted to Engineering Structures April 10, 2020 adopted subordinate systems with tuned spring-mass systems or tuned dynamic ﬂywheels. Although these subordinate-oscillators have been widely employed in engineering structures, one limitation arises from its dependence on tuning to a speciﬁc frequency. As most linear DV As are tuned to a frequency, the e ectiveness of the dynamic absorber is limited to a narrow bandwidth. This can be impractical if (i) the natural frequency of the host structure changes, for instance, with operational loads, (ii) there is a mismatch between the resonant frequency of the structure and the tuned frequency of the DV A, and (iii) the host structure is under a broadband dynamic load. To address some of these drawbacks, researchers have developed tunable dynamic absorbers, nonlinear DV As, active absorbers, and subordinate oscillator arrays (SOAs). Subordinate oscillator arrays can be viewed as an array of linear DV As that achieve vibration atten- uation over a broad range of frequencies [2–6]. The isolated natural frequencies of the oscillators in the SOA are spaced around the natural frequencies of the host structure to achieve broadband attenua- tion. Various design approaches can accomplish this goal. The most straightforward method involves the placement of the isolated natural frequencies around the host structure’s natural frequency in a random manner until the necessary attenuation is achieved over the required frequency range. A di erent ap- proach would involve adding one oscillator at a time into the array corresponding to the highest observed peak in the desired frequency range and reevaluating the frequency response of the system each time. Naturally, this is a slower approach. However, it is highly e cient, and the SOAs designed using this approach have considerable fewer number of oscillators. Vignola et al. [6] proposed a method in which discrete distributions are assigned to the natural frequencies of the oscillators. As opposed to choosing individual parameters of oscillators, assigning distributions reduces the number of parameters needed to achieve the desired frequency distribution. The performance of an SOA is highly dependent on the host and SOA uncertainty as well as the hosts structural properties. Vignola et al. [7] studied the e ect of disorder on the SOA and showed that the SOAs performance deteriorated with an increasing error in the properties. The design approaches mentioned above do not take uncertainty in structural properties into consideration during the initial design phase. Arrays with piezoelectric oscillators, studied in [8–10], can adapt their e ective structural properties in the presence of uncertainties. However, such an array of oscillators are complex to design and build when compared to ordinary SOAs. In this paper, a design approach is adapted from the second method discussed above, that takes the uncertainties of the system into consideration and corrects the model to account for the disorder in the system. The e ectiveness of this approach is tested using experiments on a two degree of freedom system, a physical model that mimics a two-story building. Fettuccine strands are used as cantilever beams for oscillators in the array. Since Young’s modulus of fettuccine can vary from strand to strand, they make a suitable and inexpensive test subject for our approach. Another objective of the present work is to bridge the gap between the research and the teaching communities via hands-on experimentation. Although many structural dynamics textbooks have dis- cussed vibration absorbers [11, 12], there is a need for pedagogical clarity so that students can have a deeper understanding and appreciation of the underlying concepts. A hands-on approach that exempli- ﬁes the nuances of experimentally realizing a vibration absorber should be part of a traditional classroom learning process in vibrations. Therefore, the focus of the current e ort is also to develop a strategy that assists in teaching principles behind vibration attenuation by extending classroom lectures through hands-on experience. It should also be noted that the novel concepts of meta-structures can also be explained via this very simple example. For instance, Inman et al. develop a set of meta-structures in [13] that are simply a more sophisticated approach to the same concept. Although meta-structures can be elusive for entry level teaching and classes, an example such as this can relate the simple concepts of DV A and extend 2 2-DOF Host Structure Host Structure with SOAs 3D Rendering of the Host Structure with SOAs (a) 2-DOF Host Structure 2-DOF Host Structure Host Structure with SOAs 3D Rendering of the Host Structure with SOAs (b) Host Structure with SOAs 2-DOF Host Structure Host Structure with SOAs 3D Rendering of the Host Structure with SOAs (c) 3D rendering of the structure Figure 1: Host Structure them to meta-structures. As a part of this goal, SOAs are developed using non-conventional, yet mundane materials such as fettuccine and modeling clay. In designing tuned oscillators from such constituents, students have to apply theoretical formulations ﬁrst to estimate approximate material properties and design oscillators to a particular frequency. The next step is to experimentally measure the oscillator’s fundamental frequency, and observe the presence of uncertainty in the boundary conditions and material properties. This gives the students a chance to further tune the frequency by either reducing the length of the pasta or by adding tip-mass. Such an innovative and engaging hands-on approach compliments some of the recent e orts by the structural community to enhance the learning experience by using 3D printed technology [14] for teaching concepts of structural dynamics [15–18] through engaging experiments. 2. Analytical model of a 2-DOF host structure This section presents the governing equations that model the 2-DOF structure shown in Figure 1. In later sections, SOAs are designed to target the natural frequencies of this representation of a two- story building. Figure 1 displays a 3D rendering of the host structure, along with a spring-mass-damper description of the structure. The 2-DOF system is modeled via a Lagrangian approach by determining the kinetic and potential energy of the host structure. Typically for a structure as in Figure 1, inertial mass is assumed to be approximately equal to the two-ﬂoor masses. Therefore, the total kinetic energy of this setup is given by T=Tmass 1+Tmass 2=1 2m1˙x2 1(t)+1 2m2˙x2 2(t); (1) where x1andx2are the displacements of the two inertial masses ( m1andm2) when its base is perturbed by a displacement z; also, (˙) represents the time derivatived dt(). The expression for the potential energy of the host structure can be expressed using U=1 2k1(x2z)2+1 2k2(x2x1)2; (2) 3 where k1andk2are the e ective sti nesses of levels 1 and 2 of the two story model, respectively. Each level is modeled as four clamped-guided beams in parallel. Thus, the e ective sti ness has the relation ki=12EI L3 iN; i=1;2and N =4; (3) where EandIare the Young’s modulus and the moment of inertia of the four supporting beams, re- spectively. Following the Lagrangian approach, the equations of motion of the host structure take the following form "m10 0m2# | {z } M(¨x1 ¨x2) +"k1k1 k1k1+k2# | {z } K(x1 x2) ="0 k2# |{z} Bn zo ; (4) whereBis the control inﬂuence matrix. Analytically, the transfer functions have the form X(˙{!) Z(˙{!)= M!2+˙{C!+K1B; (5) where the damping matrix Cis obtained from the experimental modal damping values following the procedure shown in [11]. However, if the supporting columns also contribute to the mass of the structure, the inertial contribution of the supporting columns should also be included in the mass matrix ( M). Therefore, the kinetic energy of the four columns supporting the ithstory is given by Tlevel i=N 1 2AZL2 0( i˙xi1+ i˙xi)2dy! ; (6) =1 2N" AZLi 0 2 idy# | {z } mb i1˙x2 i1+N" AZLi 0 i idy# | {z } mb i2˙xi1˙xi+1 2N" AZLi 0 2 idy# | {z } mb i3˙x2 i; =1 2mb i1˙x2 i1+mb i2˙xi1˙xi+1 2mb i3˙x2 i; i=1;2 where i=" 1(3Li2y)y2 L3 i# ; i="(3Li2y)y2 L3 i# : The total kinetic energy of the 2-DOF setup is given by T=Tmass 1+Tlevel 1+Tmass 2+Tlevel 2; =1 2m1˙x2 1(t)+1 2mb 11˙z2+mb 12˙z˙x1+1 2mb 13˙x2 1 +1 2m2˙x2 2(t)+1 2mb 21˙x2 1+mb 22˙x1˙x2+1 2mb 23˙x2 2: The expressions for the updated kinetic energy and the potential energy can be substituted into the La- grange equation to obtain the governing equations of the system. The updated equations of motion of the 4 host structure are "m11 m12 m21 m22#(¨x1 ¨x2) +"k1k1 k1k1+k2#(x1 x2) ="0 0 k2mb 12# | {z } B(z ¨z) ; (7) with m11=mb 11+m1+mb 21, m12= m21=mb 22, and m22=mb 23+m2. 3. Analytical Model of Subordinate Oscillators Arrays Dynamic vibration absorbers (DV As) are ancillary systems attached to a host structure to absorb undesired vibrations of the structure. Typically, DV As are modeled as a single degree of freedom linear oscillators, as each DV A is tuned to operate primarily at its fundamental natural frequency. When an array of linear oscillators target a frequency range, then the array is referred to as a Subordinate Oscillator Array (SOA). The performance of the SOA depends primarily on the placement of the fundamental natural frequency of each oscillators in the SOA with respect to that of the host structure. A lumped-mass model of the SOAs attached to each mass of the 2-DOF system is shown in Figure 1. The equations of motion of an SOA connected to mass ( mi) will take a form of M1j¨x1j+C1j˙x1jC1j˙x1+K1jx1jK1jx1=0; (8) M2j¨x2j+C2j˙x2jC2j˙x2+K2jx2jK2jx2=0; where x1jandx2jrepresent the vector of displacements of the oscillators in the SOAs associated to the jthnatural frequency that are attached to masses 1 ;2, respectively. In the above equations, Mi j,Ci jand Ki jrepresents mass, modal damping, and sti ness vectors of the individual oscillators attached to the primary mass iand associated with natural frequency j, respectively. If X=n xT 11;xT 12;xT 21;xT 22;x1;x2oT denotes the vector of states related to the SOAs, the governing equations can be expressed in the form M¨X+C˙X+KX=F; (9) with M=266666666666664diag(M11) diag(M22) m11 m12 m21 m22377777777777775; F=8>>>>>><>>>>>>:0 0 0 k2zmb 12¨z9>>>>>>=>>>>>>;; C=266666666666664diag(C11) C11 0 diag(C22) 0C22 C11 0 c11+PC11n c12 0C22 c21 c22+PC22n377777777777775; K=266666666666664diag(K11) K11 0 diag(K22) 0K22 K11 0 k2+PK11nk2 0K22k2 k1+k2+PK22n377777777777775: In succeeding sections, dynamic model of the host and the subordinate system are validated through experimental testing. 5 4. Testing, Parameter Estimation and Host Model Validation This section validates the host structure model via experiments. In [7], Vignola et al. explain the importance of having precise estimates of natural frequencies of the host structure for accurately design- ing SOAs. Therefore, before proceeding to the design of SOAs, it is crucial to update (if required) and validate the host structure’s model. For this reason, the 2-DOF structure shown in Figure 2 is treated as the host structure whose resonant frequencies are targeted with the dynamic oscillators. In this structure, two wooden blocks are supported by four aluminum posts that are attached to a steel base plate. The dimensions and the blueprint of the host structure are provided in the supplementary documents. Addi- tionally, some of the signiﬁcant geometric parameters of the host structure are tabulated in Tables 1 and 2. Table 1: Geometric properties of the host structure Parameter Measured Tuned Mass of ﬁrst block ( m1) 0 :259kg Mass of second block ( m2) 0 :258kg Width of Al columns 0 :5in Height of Al columns 1 =16in Dist. b /w base and m226:0350:635cm (10:250:25in) 26:48cm Dist. b /wm1andm226:0350:635cm (10:250:25in) 25:62cm Table 2: Material and dynamic properties of the host structure Parameter Estimates Tuned Experimental Error Eof Aluminum 69 GPa of Aluminum 2700 kg=m3 First natural frequency 4 :94Hz 4:76Hz 4:76Hz Second natural frequency 12 :99Hz 12:87Hz 12:87Hz Experimental frequency response functions (FRFs) are measured by exciting the base of the structure with an APS (Electro-SEIS 113) shaker, as shown in Figure 2. With accelerometers attached to the base plate and the two wooden blocks, an LMS SCADAS Mobile data acquisition system was used to measure the two transmissibility transfer functions. Subsequently, the Polymax algorithm was used to determine the natural frequencies and the corresponding modal damping ratios from the measured FRFs. The dynamic model discussed in the previous section was compared with the experimental results. Then, the column lengths in the model were updated so that the model represents the experiment more accurately. Engineering judgment plays a vital role in updating the model by selecting parameters that are the sources of uncertainty. Joints add signiﬁcant complexity in representing the dynamics of the structure as a lumped-mass system. For instance, if any one of the supporting columns is not rigidly secured, its sti ness contribution is reduced; as a result, experiments yield lower results of the host’s natural frequencies. The other source of uncertainty comes from the e ective length of the supporting column between any two levels. Although the length of the support structure a ects both the mass and the sti ness matrices, its contribution to the sti ness matrix is dominant. This is due to the cubic relationship in Eq. 3. 6 AMPLIFIERACCELEROMETER OUTPUT STEPPED SINE INPUT APS SKAKERAcc.1 Acc.2 Ref. Acc. LMS DAQ Front View Side View DVA -1a DVA -1b DVA -1c DVA -2a DVA -2b DVA -2cMass 1 Mass 2 Figure 2: Host Structure Setup In a classroom, such uncertainties in the experimental setup provide an opportunity for students to develop intuition through trial and error. In the experimental setup, the mass of the two wooden blocks and the cross-sectional dimensions of the beam can be measured. However, in the current setup, the uncertainty in measuring the e ective length of the supporting posts comes from the thickness of the two wooden blocks attached to it. Therefore, the length of the posts is one parameter that is updated in the model by deﬁning a cost function in terms of experimental and simulated natural frequencies: J1=jj!exp;1!model;1(L1;L2)jj2+jj!exp;2!model;2(L1;L2)jj2: (10) The column lengths L1andL2that minimize the above cost function are the e ective lengths of the posts. An iterative approach is adopted to estimate the optimal lengths by searching in the range of uncertainty. Finally, the updated properties of the host structure are tabulated in Table 1. Also, the modal damping estimates used in the simulations are 0 :2% and 0:35% for the ﬁrst and the second models, respectively. Table 2 summarizes the characteristics of the dynamic model before and after updating the equivalent sti ness and modal damping. 5. Testing, Parameter Estimation and SOA Design This section discusses the design of SOAs to attenuate the dynamics of the 2-DOF structure shown in Figure 2. These SOAs are developed using fettuccine strands and modeling clay. While the cantilevered fettuccine strand predominately contributes to the sti ness of each oscillator, the modeling clay attached to its tip allows for ﬁne-tuning of its fundamental frequency. In a classroom, fettuccine and modeling clay are excellent and a ordable candidates for teaching the concepts of vibration absorption as they capture students’ attention and keep them engaged throughout the experimentation. As the chosen materials are unconventional candidates, the material properties are unknown and non-homogeneous, and therefore, the analysis of the system has multiple levels of complexity. Additionally, the variation in the dynamic properties between fettuccine strands gives students an opportunity to deal with uncertainty. Once the nominal material properties are estimated, the next step is to design an oscillator tuned to a particular frequency. If an oscillator is tuned to the natural frequency of a host structure, the correspond- ing peak in the frequency response function (FRF) splits into two newer peaks, and an anti-resonant 7 4 6 8 10 12 14 Frequency (Hz)-60-40-200204060Magnitude (dB)FRF of Mass 1 Experiment Updated Model Initial Model 4 6 8 10 12 14 Frequency (Hz)-60-40-200204060Magnitude (dB)FRF of Mass 2 Experiment Updated Model Initial ModelFigure 3: Experimental and simulated FRFs of the host structure. valley appears at the tuned frequency. A DV A is an example of such an oscillator which, when tuned to the host’s natural frequency, absorbs the host’s energy and creates a new FRF with two peaks. However, the dynamics of the system can deteriorate when the modiﬁed structure is under broadband excitation, thereby activating the two newer resonant peaks. As a result, the natural progression of this discus- sion seeks an array of oscillators (SOAs) that attenuates the dynamic response over a broader frequency bandwidth. Designing and implementing SOAs using fettuccine strands in a classroom is very engaging as it is highly intuitive to tune oscillators. However, students would realize that for precise tuning of each oscillator, it is necessary to estimate the accurate dynamic properties of individual fettuccine strands. In this regard, towards the end of this section, an iterative process is proposed to develop better performing SOAs under the presence of uncertainties. The rest of this section provides details of each of these steps along with corresponding experimental results. 5.1. Estimation of material properties of Fettuccine strands In this section, the average Young’s modulus of oscillators is experimentally estimated using the dynamics test setup shown in Figure 4b. The tip response of the cantilevered fettuccine strand (without any tip-mass) is measured with a PSV-100 laser vibrometer as the base is mounted to an APS shaker. The resonant frequency of this strand is then determined from the experimental FRF. The Young’s modulus of the fettuccine strands are estimated from the measured resonant frequency ( !) and the average material 8 density () using the analytical relationship [11] given by !=s 3EI (0:24AL)L3; (11) where the geometric parameters are length ( L), cross-sectional area ( A) and moment-of-inertia ( I). Ta- ble 3 presents the average structural parameters of fettuccine strands. Figure 4a shows the frequency response function of a fettuccine strand as its length is varied, and Table 3 lists the corresponding nat- ural frequencies and the damping ratios. Based on these experimental results, one can see that Young’s modulus of the fettuccine strand varies from 2 :14109Pato 2:63109Pa. Since the other parameters do not show as signiﬁcant a variation as the Young’s Modulus, uncertainties in the FRFs can be primarily attributed to the variance in Young’s modulus. 5 10 15 20 25 Frequency (Hz)-60-50-40-30-20-100Magnitude (dB)8 in 7 in 6 in 5 in (a) (b) Figure 4: (a) Experimental FRFs of fettuccine strands of varying lengths (Input: acceleration of the base, Output: Tip-velocity of the fettuccine strand), (b) Dynamic test of a single cantilevered fettuccine strand Table 3: Measured and estimated physical parameters of fettuccine strands DV A length Exp.  Exp. Estimated E Parameter Value 8in 8.35045 Hz 1.76494% 2.5851 GPa Average Thickness 1:4mm 7in 11.0095 Hz 1.8529% 2.634 GPa Average Width 5:6mm 6in 13.5171 Hz 1.89777% 2.1432 GPa Average Density 1115:8kg=m3 5in 19.7543 Hz 2.31913% 2.2075 GPa Area Moment of Inertia 1:29e12m4 =1.9587% =2.3925 GPa is mean estimate, - natural frequency, - modal damping ratio and E- Young’s modulus 5.2. Numerical simulation of SOAs Before proceeding to the experimental implementation of SOAs, the dynamics of the SOAs are simu- lated using an array of identical fettuccine strands with Young’s modulus and modal damping ratio equal to the experimental mean estimate of 2 :39GPa and 1:95 %, respectively. The other parameters of the oscillators are presented in Table 4. 9 4 4.5 5 5.5 Frequency (Hz)0204060Magnitude (dB)FRF of Mass 1 Host 1 SOA 2 SOA 3 SOA 4 4.5 5 5.5 Frequency (Hz)0204060Magnitude (dB)FRF of Mass 2 Host 1 SOA 2 SOA 3 SOA(a) 1212.5 1313.5 14 Frequency (Hz)0102030Magnitude (dB)FRF of Mass 1 Host 1 SOA 2 SOA 3 SOA 1212.5 1313.5 14 Frequency (Hz)0102030Magnitude (dB)FRF of Mass 2 Host 1 SOA 2 SOA 3 SOA (b) Figure 5: Simulated FRFs of the host structure as oscillators are added to build the SOA at (a) the ﬁrst resonant peak, and (b) the second resonant peak Figure 5 shows the FRF of the host structure with and without oscillators. As oscillators are added one at a time, the magnitude of the FRF is reducing over a broader frequency bandwidth. SOAs with three oscillators is able to ﬂatten the FRF over the frequency bandwidth of interest. Typically, a tuned oscillator targets the peak frequency of the system. However, sometimes, due to practical limitations, they are mistuned, i.e., the peak frequency and the target frequency are not equal to each other. Table 4 presents the selected target and peak frequencies of all oscillators in the two SOAs. The peak’s magnitude and the vibration attenuation achieved in each iteration is also shown in Figure 5. After addition of three oscillators, the (minimum) vibration attenuation achieved are approximately 25 dBand 18 dBat the ﬁrst and second natural frequency bands, respectively. Figure 5 also shows that the addition of SOAs has resulted in modal spillover. In other words, there is a minimal increase in magnitude at some of the natural frequencies around the host’s initial peak. However, this increase in magnitude is negligible when compared to the extent of attenuation achieved by using SOAs. As seen in Figure 5a, oscillator OSC-1a tuned to peak frequency splits the resonant peak into two peaks of equal magnitudes. However, in Figure 5b oscillator OSC-2a is mistuned and as a result, it splits the resonant peak into two peaks of di ering magnitudes. Similar behavior is observed with other oscillators in the two SOAs. It is also interesting to note that with the current approach targeting peak frequencies, even number of oscillators or mistuned oscillators in the SOA results in FRFs that have 10 Table 4: Peak magnitude and attenuation achieved by SOAs as observed in numerical simulations. Natural Freq. Oscillator (OSC)Frequency Max Peak Magnitude Max Peak Attenuation (SOA) Max Peak Target Mass 1 Mass 2 Mass 1 Mass 2 1stNo OSC 4:76Hz 4:76Hz 45:6dB 49:29dB - - OSC - 1a 4:92Hz 4:92Hz 28:25dB 31:95dB 17:35dB 17:34dB (SOA-1)OSC - 1a, 1b 4:56Hz 4:58Hz 26:37dB 30:01dB 19:23dB 19:28dB OSC - 1a, 1b, 1c 4:65Hz - 21:03dB 24:65dB 24:57dB 24:64dB 2ndNo OSC 12:86Hz 12:76Hz 31:09dB 28:14dB - - OSC - 2a 12:5Hz 12:55Hz 18:81dB 16:37dB 12:28dB 11:77dB (SOA-2)OSC - 2a, 2b 13:20Hz 13:20Hz 17:84dB 15:6dB 13:25dB 12:54dB OSC - 2a, 2b, 2c 12:2Hz - 12:95dB 10:3dB 18:14dB 17:84dB uneven peak magnitudes. Thus, the question arises whether SOAs can be implemented practically and the next subsection discusses this question using experimental results. 5.3. Designing SOAs using oscillators with uncertain material properties This section studies the performance of SOAs when the variation in Young’s modulus of the fettuccine strands is neglected and oscillators based on nominal material properties estimated in the previous section are used to construct the SOAs. Fettuccine based oscillators are added to the host structure one at a time, and the tip-mass is calculated using the nominal Young’s modulus to tune the oscillator to the peak frequency. However, due to the uncertainty, the oscillator’s natural frequency may not match the target frequency. Rather than updating the tip-mass, a new oscillator is added, and the process is repeated until a ﬂat FRF is achieved. The details of the tip-masses, the lengths of the fettuccine stands and the target frequencies ( ) are tabulated for the two SOAs in Table 5. Eventually, two SOAs of nine and six oscillators resulted in a ﬂat FRF at the ﬁrst and the second resonant frequencies of the host structure. SOA-1 and SOA-2 are attached to the top and the bottom wooden blocks, respectively as shown in Figure 6b. Figure 6a shows the FRF of the modiﬁed structure with SOAs. This ﬁgure shows that with SOAs, the dynamic response of the host structure can be attenuated over a broad frequency bandwidth in spite of uncertain material properties. However, the attenuated response is uneven with a nonuniform magnitude in its frequency bandwidth. If precise dynamic properties of the oscillators are known, the resulting attenuated response will result in uniform magnitude. Also, a comparable degree of attenuation could be achieved with a fewer number of oscillators. In the next section, an iterative approach is proposed to update fettuccine’s material properties, thereby reducing the uncertainties while designing SOAs. 5.4. Designing SOAs using oscillators by updating material properties The ﬂowchart in Figure 7 presents a generic approach to design SOAs in the presence of uncertainties. The design approach is primarily based on the idea of estimating the natural frequency of a substructure after it is attached to a host /primary structure. In this paper, the analysis is restricted to a simpliﬁed method when the approximate range of variation of the parameter is known. From preliminary experi- ments conducted, it is observed that the uncertainty in the substructure arose primarily due to variance in Young’s modulus of the material. The approximate variation of the Young’s Modulus assessed in previ- ous sections is used here. Furthermore, this method inherently assumes that the mass of each subordinate oscillator is negligible when compared to that of the host structure. The ﬁrst step in the design process involves the selection of a frequency band with a resonant peak. Then a single oscillator is attached such that its natural frequency is higher than the selected frequency 11 4 6 8 10 12 14 Frequency (Hz)-60-40-200204060Magnitude (dB)FRF of Mass 1 No SOA With SOA 4 6 8 10 12 14 Frequency (Hz)-60-40-200204060Magnitude (dB)FRF of Mass 2 No SOA With SOA(a) (b) Figure 6: Vibration attenuation using SOAs made of oscillators with unknown material properties Table 5: Properties and target frequencies of the oscillators with uncertain Young’s modulus that achieved ﬂat band attenuation of the host SOA-1 Oscillator OSC-1a OSC-1b OSC-1c OSC-1d OSC-1e OSC-1f OSC-1h OSC-1h OSC-1i Length 8in 8in 8in 8in 8in 8in 8in 8in 8in Tip-mass 0:80g 1:00g 1:03g 0:83g 1:27g 1:35g 1:36g 0:71g 0:65g Target 4:78Hz 4:70Hz 4:60Hz 5:04Hz 4:47Hz 4:39Hz 4:4Hz 5:21Hz 5:3Hz SOA-2 Oscillator OSC-2a OSC-2b OSC-2c OSC-2d OSC-2e OSC-2f Length 5in 5in 5in 5in 5in 5in Target 12:98Hz 13:14Hz 12:47Hz 12:31Hz 13:53Hz 12:17Hz band. The assumptions mentioned earlier become relevant in this step. After attaching the fettuccine strand (without tip-mass) to the host structure, the e ective natural frequency of the substructure can be estimated by placing an accelerometer at the base of the oscillator and measuring its tip response. Even 12 Determine Natural Frequency Attach new oscillator without tip massDetermine Natural Frequency of the new oscillator Estimate Y oung's Modulus of the oscillator Calculate tip- mass of the oscillatordB reduction achieved?YesHost FRF Attenuated structureEstimate response of the structure with all oscillators in the SOANoFigure 7: Flow chart of the iterative procedure followed to estimate material properties of fettuccine strands and therefore design oscillators tuned to the desired target frequency. though the substructure is attached to the host, the natural frequency of the substructure can be identiﬁed from the frequency response of the substructure measured between its base and its tip. However, in cases where the dynamics of the oscillator cannot be directly measured, the FRF between the excitation and the response of the host structure is su cient. For simple structures such as the one under study, care is taken such that the natural frequency of the fettuccine strands (without tip-mass) is su ciently away from the two frequency bandwidth of interest. In such a scenario, a new peak is introduced in the frequency response function of the host structure. For instance, an untuned fettuccine strand of 8 in has natural frequency around 8.5 Hz, which does not lie in the two frequency bandwidths of interest [ 4-5.5 ] Hzand [ 12-14 ] Hz. Due to the low mass-ratio as well the fact that the natural frequency of the substructure is signiﬁcantly di erent from that of the host, the frequency corresponding to the induced peak can be considered as a good approximation of the substructure’s natural frequency. This is then tuned to target the peak at 4.76 Hzby attaching a calculated tip-mass of 0.914 g. Table 4 has further details of all oscillators, where the natural frequencies of the oscillators are estimated using Polymax algorithm. A precise estimate of Young’s modulus of each fettuccine strand is calculated using Eq. 11. Table 6: Estimated dynamic characteristics of the individual oscillators OSC parameter estimation OSC tuning step OSC no. Length Exp. Exp. Estimated E TargetTip-mass Calculated Tuned ErrorSOA-1OSC-1a 8 in 8.46 Hz 1.36% 2.57 GPa 4:76Hz 0:914g 0:930g1.7% OSC-1b 8 in 8.56 Hz 1.03% 2.63 GPa 4:92Hz 0:819g 0:840g2.5% OSC-1c 8 in 8.05 Hz 0.87% 2.39 GPa 4:58Hz 0:966g 0:970g0.4%SOA-2OSC-2a 5 in 20.67 Hz 2.04% 2.34 GPa 12:76Hz 0:416g 0:440g5.7% OSC-2b 5 in 20.73 Hz 1.84% 2.35 GPa 12:55Hz 0:460g 0:480g4.3% OSC-2c 5 in 20:22Hz 2.05% 2.24 GPa 13:20Hz 0:347g 0:350g0.8% =1.53%=2.42 GPa is the mean estimate, - natural frequency, - modal damping ratio and E- Young’s modulus Once the precise ﬂexural sti ness is estimated, the natural step is to determine the tip-mass to tune the oscillator. Previously, the tip-mass is estimated based on the peaks in the FRFs. The magnitudes of 13 4.24.44.64.855.25.4 Frequency (Hz)0204060Magnitude (dB)FRF of Mass 1 Host 1 SOA 2 SOA 3 SOA 4.24.44.64.855.25.4 Frequency (Hz)0204060Magnitude (dB)FRF of Mass 2 Host 1 SOA 2 SOA 3 SOA(a) 1212.5 1313.5 14 Frequency (Hz)0102030Magnitude (dB)FRF of Mass 1 Host 1 SOA 2 SOA 3 SOA 1212.5 1313.5 14 Frequency (Hz)0102030Magnitude (dB)FRF of Mass 2 Host 1 SOA 2 SOA 3 SOA (b) Figure 8: Experimental Iterative FRFs of SOAs Table 7: Peak magnitude and attenuation achieved during each step of iterative SOA design while following the proposed design methodology Natural Freq. OSC-No.Frequency Peak’s Magnitude Attenuation Peak’s Target’s Mass 1 Mass 2 Mass 1 Mass 2 1stNo OSC 4:76Hz 4:76Hz 49:17dB 45:48dB - - OSC - 1a 4:92Hz 4:92Hz 30:47dB 26:81dB 18:6dB 18:67dB OSC - 1a, 1b 4:56Hz 4:58Hz 31:59dB 27:79dB 17:58dB 17:69dB OSC - 1a, 1b, 1c 4:65Hz - 23:91dB 20:15dB 25:26dB 25:33dB 2ndNo OSC 12:86Hz 12:76Hz 28:46dB 31:13dB - - OSC - 2a 12:5Hz 12:55Hz 14:61dB 16:49dB 13:85dB 14:64dB OSC - 2a, 2b 13:20Hz 13:20Hz 11:59dB 15:17dB 16:87dB 15:96dB OSC - 2a, 2b, 2c 12:2Hz - 11:19dB 12:23dB 17:27dB 18:9dB the two new peaks may not be equal, as natural frequencies of the oscillators in an SOA are very close to each other. Alternatively, the tip-mass is estimated such that the target peak splits into two peaks of equal magnitudes. The analytical model of the 2-DOF system with oscillators is used to iterate through a set of 14 4 6 8 10 12 14 Frequency (Hz)-60-40-200204060Magnitude (dB)FRF of Mass 1 Host With SOA 4 6 8 10 12 14 Frequency (Hz)-60-40-200204060Magnitude (dB)FRF of Mass 2 Host With SOAFigure 9: Complete experimental FRF of the host with SOAs made of fettuccine and modeling clay permissible tip-masses. Consequently, the tip-mass resulting in two peaks of equal magnitudes is added to the oscillator. The above procedure is repeated until the desired reduction in magnitude, or ﬂat FRF is achieved. This design methodology is implemented on the current test-setup, and the corresponding experi- mental results are presented in Figure 8. As discussed earlier, the fettuccine oscillators are ﬁrst attached (without tip-mass) to the host to estimate its modal damping and Young’s modulus. Table 6 summarizes these estimates along with the tip-mass calculated from simulations. This table also compares the cal- culated tip-mass value with the ﬁnal attached clay mass that resulted in peaks of equal magnitude in the FRF. This discrepancy in the theoretical and the actual tip-mass values arises because a point mass does not accurately model the clay at the tip of the cantilevered strand. Table 7 summarizes the peak magnitudes of the modiﬁed structure along with the attenuation achieved at each step of the iteration. Additionally, Figure 6 shows the progression of the FRFs as the host is mod- iﬁed. This ﬁgure shows that tuning SOAs of three oscillators can experimentally create a ﬂat spectrum similar to the simulation results shown in Figure 5. Also, having knowledge of the precise dynamic properties of the oscillators is important to get reasonably uniform peaks in the frequency bandwidth of interest. In the previous section, 28.55 dB and 20.24 dB reduction in magnitude of the mass 1’s (ﬁrst and second) natural frequencies were achieved with SOAs of nine and six mistuned oscillators, respectively. However, by precisely tuning the oscillators, 25.26 dB and 17.27 dB reduction is achieved for the same 15 with two SOAs of three oscillators only. The di erence in the quality of the ﬁnal response is evident from Figure 6a and Figure 9. Table 8 summarizes the minimum magnitude reduction achieved around the ﬁrst and second natural frequencies for the 2-DOF system. Also, when a single oscillator case is compared against SOAs, SOAs show a better broadband performance than a single oscillator. The table also in- dicates that the e ectiveness of the SOA designed using random placement methods is only marginally better than that of the SOA designed using the proposed approach. This di erence can be attributed to the much higher number of oscillators in the SOA designed using random placement method. One can argue the validity of the comparisons between the SOAs and the single oscillator since their mass ratios di er vastly. This shows that mass-ratio is one of the important parameters, and in the next section SOAs are compared against a single oscillator with matching mass-ratio. Table 8: Comparison of performance of single oscillator and SOAs designed using random placement and proposed design methods Natural Freq. AbsorberMagnitude Reduction across BandMass RatioMass 1 Mass 2 1stSingle oscillator (OSC-1a) 18:6dB 18:67dB 0:4% SOA - random placement 28:55dB 28:5dB 3:9% SOA - proposed method 25:26dB 25:33dB 1:2% 2ndSingle oscillator (OSC-2a) 13:85dB 14:64dB 0:2% SOA - random placement 20:239dB 21:325dB 1:5% SOA - proposed method 17:27dB 18:9dB 0:7% Table 9: Magnitude reduction at peak’s frequency at di erent damping ratios of the oscillators in the SOA Natural Freq. Damping RatioPeak Magnitude Peak Frequency Reduction at Peak’s Frequency Mass 1 Mass 2 Mass 1 Mass 2 Mass 1 Mass 2 1stHost 49:17dB 45:48dB 4:76Hz 4:76Hz - - 0:1% 42:78dB 38:86dB 4:471Hz 4:471Hz 6:39dB 6:62dB 0:5% 32:96dB 29:1dB 4:469Hz 4:469Hz 16:21dB 16:38dB 1% 28:39dB 24:63dB 4:466Hz 4:466Hz 20:78dB 20:85dB 1:96% 24:86dB 21:25dB 4:466Hz 4:466Hz 24:31dB 24:23dB 3% 23:99dB 20:35dB 4:677Hz 4:668Hz 25:18dB 25:13dB 2ndHost 28:46dB 31:13dB 12:86Hz 12:86Hz - - 0:1% 24:34dB 27:48dB 12:3Hz 13:41Hz 4:12dB 3:65dB 0:5% 16:53dB 19:98dB 12:29Hz 13:42Hz 11:93dB 11:15dB 1% 12:87dB 16:04dB 12:28Hz 13:43Hz 15:59dB 15:09dB 1:96% 10:3dB 12:94dB 12:3Hz 13:42Hz 18:16dB 18:19dB 3% 9:874dB 12:3dB 12:59Hz 12:69Hz 18:586dB 18:83dB 6. Parametric Study 6.1. E ect of damping One of the crucial factors that deﬁne the ﬂatness of the frequency band is damping of the subordinate oscillators. As evident from Figure 10, the peaks, and valleys in the frequency band disappear as the damping ratio reaches 1 :96%, and thus a ﬂatter frequency response is achieved. Note that the value 16 1:96% is the average damping ratio obtained from the preliminary fettuccine experiments. Table 9 lists the amount of magnitude reduction achieved at di erent damping ratios. As evident from the ﬁgure and the table, increasing the damping ratio reduces the magnitude peaks across the band. However, beyond a critical damping value, which is 1 :96% in this case, the magnitude reduction is no longer signiﬁcant. This analysis proves that the damping ratio and hence the material chosen for the oscillators can inﬂuence the SOA’s performance signiﬁcantly. In cases where one has to select oscillators with low damping ratios for di erent design situations, SOAs with a higher number of oscillators can produce the desired frequency response [9]. (a) First Resonant Frequency (b) Second Resonant Frequency Figure 10: E ect of damping on the FRFs of the host with SOAs 6.2. Robustness Study Previous sections highlighted that SOAs are useful for broadband vibration attenuation, and it is known that single oscillator absorbers /DV As are meant for cases where the frequency of excitation is rel- atively constant. In this section, the performance of SOAs and DV As are investigated when the dynamic properties of the host are perturbed. The focus of this study is to investigate the performance of DV As and SOAs when the natural frequency shifts from !to ˜!. Figures 11 and 12 compare the robustness of the SOA with that of the DV A. These ﬁgures present two ways of comparing the performance of SOAs and DV As when the host’s natural frequency is varied by 10%, i.e., 0 :9˜!=!1:1. The x-axis of 17 these ﬁgures is the perturbed non-dimensional natural frequency (i.e. ˜ !=!), and the value of 1 on the x-axis corresponds to the unperturbed case discussed in the previous sections. In Figure 11, the dBreduction is calculated between the peak magnitude of the unmodiﬁed host and the highest peak’s magnitude of the modiﬁed /perturbed host. For the ﬁrst and the second resonant frequency,!is equal to 4 :76Hzand 12:86Hz, respectively, and the dBreduction in the ﬁgure gives a measure of the minimum attenuation achieved around the corresponding host’s resonant peaks. One can observe that the dBreduction in the case of SOA is higher than DV A at ˜ !=! =1 ( the unperturbed case ), which conforms with the previous discussions and intuition. As the host’s properties are perturbed, the performance of both the systems deteriorate. However, even at high levels of perturbation, the SOA’s performance is always better than that of the DV A for 0 :9˜!=!1:1. The shaded region shown in the ﬁgure gives a ﬁgurative intuition of the level of perturbation required for the system with SOA to deteriorate to the best performance achieved by the DV A. As DV As have the best performance at their tuned frequency - !, Figure 12 presents the dBreduction at!. Given that the host’s natural frequency is perturbed to ˜ !, the DV A is mistuned and its performance deteriorates drastically as seen in this ﬁgure. However, SOA’s magnitude reduction remains fairly con- stant over a higher perturbation. Beyond a critical perturbation, both the systems deteriorate drastically. It is easy to infer from both the ﬁgures that SOAs are better for structures with higher uncertainty. On the other hand, DV As are a better choice only when the uncertainty in structural parameters of the host is almost insigniﬁcant or when the system is excited at a single natural frequency, which are highly unlikely in practical applications. 7. Conclusion The present work aims to design a hands-on experiment that extends the teaching of broadband vibra- tion attenuation to a class focused on structural dynamics. An array of dynamic vibration absorbers, also called Subordinate Oscillator Arrays, are developed using fettuccine pasta and modeling clay to extend the concept and even provide simpliﬁed exposure to modern day meta-structures. The SOAs are designed to attenuate the dynamics of a 2-DOF structure. The dynamics of the modiﬁed structure is studied via simulations and experimental investigation. Acknowledgment Tarazaga would like to acknowledge the support provided by the John R. Jones III Faculty Fellowship.