Effects of Structural Parameters on the Dynamics of a Beam Structure with a Beam-Type Vibration Absorber

Abstract

A beam-type absorber has been known as one of the dynamic vibration absorbers used to suppress excessive vibration of an engineering structure. This paper studies an absorbing beam which is attached through a visco-elastic layer on a primary beam structure. Solutions of the dynamic response are presented at the midspan of the primary and absorbing beams in simply supported edges subjected to a stationary harmonic load. The effect of structural parameters, namely, rigidity ratio, mass ratio, and damping of the layer and the structure as well as the layer stiffness on the response is investigated to reduce the vibration amplitude at the fundamental frequency of the original single primary beam. It is found that this can considerably reduce the amplitude at the corresponding troublesome frequency, but compromised situation should be noted by controlling the structural parameters. The model is also validated with measured data with reasonable agreement.

1. Introduction

A beam-type absorber is one of the techniques to reduce undesirable vibration of many vibrating systems, such as a synchronous machine, mounting structure for a sensitive instrument, and other continuous structure in engineering. The absorber system usually consists of a beam attached to the host structure using an elastic element. The natural frequency of the absorber is then tuned to be the same as the troublesome operating frequency of the host structure to create counter force, which in return reduces the vibration of the structure. As beams are important structures in civil or mechanical engineering, several works have also been established to investigate the performance of the absorbing beam which is attached also to a beam structure.

Among the earliest studies of the double-beam system is one proposed by Yamaguchi [1], which investigated the effectiveness of the dynamic vibration absorber consisting of double-cantilever visco-elastic beam connected by spring and viscous damper. The auxiliary beam is attached to the center of the main beam excited at its end by a sinusoidal force. It is found that the amplitude at resonances of the main beam is sensitive to the stiffness and mass of the absorbing beam. The damping ratio was formulated as a function of mass and layer stiffness of the absorber. Vu et al. [2] studied the distributed vibration absorber under stationary distributed force. A closed form was developed by utilizing change of variables and modal analysis to decouple and solve differential equations. Oniszczuk [3] studied the free vibrations of two identical parallel simply supported beams continuously joined by an elastic layer. The eigen frequencies and mode shapes of vibration of the double-beam system were found using the classical assumed mode summation.

Another theoretical study for finding the optimum design of beam-type absorber was presented by Aida etal. [4]. A uniform absorbing beam with the same boundary condition is connected to the main beam by spring and damper components. In the study, an optimum tuning method to reduce the main beam vibration was proposed. Another structural analysis and optimum design of a dynamic absorbing beam with free edges was studied by Chen and Lin [5]. The effect of mass ratio and the stiffness layer on the vibration response was discussed.

The effect of different forcing types on the double-beam system has also been discussed by several authors. Zhang et al. [6] studied vibration characteristics of the double-beam system under axial compressive load. The studies were limited for two identical simply supported double-beam system and the effect of the axial load on the beams vibration amplitude was reported. Abu-Hilal [7] studied the effect of a moving constant load on the dynamic response of the beams. This was done for different values of speed parameter, damping ratio, and stiffness parameter.

Several works focus on the development of mathematical model to provide solutions of the vibration response. De Rosa and Lippiello [8] studied free vibration of double-identical beam system using the Differential Quadrature Method (DQM). Sadek et al. [9] presented a computational method for solving optimal control of transverse vibration of a two-parallel-beam system based on legendry wavelets approach. It is found here that the reduction of the beam vibration depends on the spring location on the beam.

This paper investigates the effect of structural parameters, namely, the rigidity ratio, mass ratio, damping loss factor, the stiffness, as well as the damping ratio of the layer on the dynamic response of simply supported double-beam system to provide a thorough analysis. The discussion is limited on controlling the fundamental mode of a single-beam structure using a dynamic absorbing beam. The following section discusses the derivation of the mathematical model.

2. Mathematical Modeling

The schematic diagram of a beam connected with an absorbing beam having the same length L and simply supported is shown in Figure 1(a). Here distributed lumped system of two degrees of freedom is assumed, where the visco-elastic element between the beams consists of parallel distributed springs and dampers as shown in Figure 1(b).

(a)

(b)

Figure 1 

(a) Schematic diagram of a visco-elastically connected simply supported double-beam system and (b) the two-degree-of-freedom system of distributed lumped parameter model.

The equation of motion of the dynamic model can therefore be written as 𝐸₁𝐼₁𝜕⁴𝑤₁𝜕𝑥⁴+𝑚₁𝑤̈₁+𝑐𝑤̇₁−𝑤̇₂+𝑘𝑤₁−𝑤₂=𝑃𝑒^𝑖𝜔𝑡,𝐸₂𝐼₂𝜕⁴𝑤₂𝜕𝑥⁴+𝑚₂𝑤̈₂+𝑐𝑤̇₂−𝑤̇₁+𝑘𝑤₂−𝑤₁=0, (1)(2) where 𝑃=𝐹₀𝛿(𝑥−𝐿/2)is the external point force with frequency 𝜔 at the midspan of the beam, 𝐹₀ is the force magnitude, 𝐾= 𝐸𝐼is the flexural rigidity, 𝑐is the damping constant, 𝑘is the stiffness constant of the viscous layer, and 𝑚is the mass of the beam. The subscripts 1 and 2 refer to the main beam and the absorber beam, respectively. The damping of the beam can be introduced by replacing the flexural rigidity in (1) and (2) by 𝐾(1+𝑖𝜂), where 𝜂 is the damping loss factor. The vertical displacement of the main beam 𝑤₁and the absorber 𝑤₂can be expressed as a series expansion in terms of mode shape function 𝑋_𝑛(𝑥) for the 𝑛th mode.The amplitudes in generalized time coordinates 𝑞_1𝑛 are given by 𝑤₁(𝑥,𝑡)=^∞_𝑛=1𝑋_𝑛(𝑥)𝑞_1𝑛(𝑡),𝑤₂(𝑥,𝑡)=^∞_𝑛=1𝑋_𝑛(𝑥)𝑞_2𝑛(𝑡), (3)(4) where 𝑥is the position on the beam at which the load is applied. For the simply supported boundary condition, the nth mode shape function can be written as 𝑋_𝑛(𝑥)=sin𝜎_𝑛𝑥, (5) where 𝜎_𝑛 is the eigenvalue of the mode shape which can be expressed as: 𝜎_𝑛=𝑛𝜋𝐿. (6) The orthogonality conditions can be used for simplifying the equations of motion which is represented in the following form [1]: ^𝐿₀𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)𝑑𝑥=0,𝑛≠𝑚. (7) Equations (3) and (4) can be substituted to (1) and (2). Multiplying both sides of the equations by mode shape function 𝑋_𝑚then integrating through the beam length and applying the orthogonality condition as in (7) yield a matrix form for the equations of motion expressed as (see Appendix) 𝑚₁00𝜇𝑚₁𝑞̈₁𝑞̈₂+𝑐−𝑐𝑞̇−𝑐𝑐𝑞̇¹₂+𝐾₁𝜎⁴_𝑛+𝑘−𝑘−𝑘𝑒𝐾₁𝜎⁴_𝑛𝑞+𝑘𝑞¹₂=𝐹₀02𝐿sin𝑛𝜋2𝑒^𝑖𝜔𝑡, (8) where 𝜇=𝑚₂/𝑚₁ is the mass ratio and 𝑒=𝐸₂𝐼₂/𝐸₁𝐼₁ is the rigidity ratio. The damping of the layer between the main beam and the absorbing beam can be approached by using the concept of “mixed damping ratio” in discrete dynamic vibration absorber [10]. The damping ratio of the visco-elastic layer can be defined as 𝑐𝜉=2𝑚₂𝜔_𝑛=𝑐2𝜇𝑚₁𝜔_𝑛, (9) where 𝜔_𝑛is the original natural frequency of the primary beam (without the absorber attached) which is given by 𝜔_𝑛=𝐸₁𝐼₁𝜎⁴_𝑛𝑚₁. (10) For a stationary harmonic load, it is therefore necessary to analyze the system performance in the frequency domain to clearly identify the distinct responses in the resonant frequencies. By defining the amplitude 𝑞 in terms of the complex exponential notation it gives 𝑞₁𝑞₂𝑄=Re¹𝑄₂𝑒^𝑖𝜔𝑡, (11) where 𝑄 is the complex amplitude of 𝑞.

Substituting (11) into (8) yields𝐾₁𝜎⁴_𝑛+𝑘+𝑖𝜔𝑐−𝜔²𝑚₁−𝑖𝜔𝑐−𝑘−𝑖𝜔𝑐−𝑘e𝐾₁𝜎⁴_𝑛+𝑘+𝑖𝜔𝑐−𝜔²𝜇𝑚₁𝑄₁𝑄₂=𝐹₀02𝐿sin𝑛𝜋2, (12) where the solutions for the complex amplitude of the frequency response function in terms of the receptance, that is, 𝑄/𝐹₀ can then be obtained for each mode 𝑛 which are given by𝑄₁𝐹₀=2e𝐾₁𝜎⁴_𝑛+𝑘+𝑖𝜔𝑐−𝜔²𝜇𝑚₁sin(𝑛𝜋/2)𝐾₁𝜎⁴_𝑛+𝑘+𝑖𝜔𝑐−𝜔²𝑚₁e𝐾₁𝜎⁴_𝑛+𝑘+𝑖𝜔𝑐−𝜔²𝜇𝑚₁−(𝑖𝜔𝑐+𝑘)²𝐿,𝑄₂𝐹₀=2(𝑖𝜔𝑐+𝑘)sin(𝑛𝜋/2)𝐾₁𝜎⁴_𝑛+𝑘+𝑖𝜔𝑐−𝜔²𝑚₁𝑒𝐾₁𝜎⁴_𝑛+𝑘+𝑖𝜔𝑐−𝜔²𝜇𝑚₁−(𝑖𝜔𝑐+𝑘)²𝐿. (13)(14)

As observed from (13) and (14), the double-beam structure is expected to have two natural frequencies for each mode of vibration, as the system is modeled using two-degree-of-freedom system. It can also be seen in (13), where for the lumped parameter system with 𝐾₁=0 and for an undamped case where c = 0, the magnitude of the numerator is proportional to 𝑘−𝜔²𝑚₂. The main beam amplitude can therefore be suppressed to zero by tuning the layer stiffness and the mass of the absorbing beam to be equal to the forcing frequency, that is, 𝜔=𝑘/𝑚₂. The subsequent sections discuss the effect of structural parameters on the response in (13) and (14).

3. Effects of Rigidity and Mass Ratio

In this investigation, the rigidity ratio and mass ratio are varied to observe their effects on the main beam response. This is calculated for a steel beam (Young’s modulus 𝐸=2.1×10¹¹ Pa, density 𝜌=7800 kg/m³) having length 2 m, width 0.065 m, and thickness 0.02 m. The stiffness of the layer is assumed 100 kN/m. The calculation here is done for the first mode of vibration (𝑛=1) for each beam. Figure 2 shows the vibration amplitude for the first two resonances of the primary beam plotted in logarithmic scale. Here the damping ratio of the visco-elastic layer and the structural damping loss factor of the beams are assumed very small, that is, 𝜁=𝜂=0.01.

(a)

(b)

(c)

(d)

Figure 2 

Effect of the rigidity ratio 𝑒 and the mass ratio 𝜇 on the vibration amplitude at the midspan location of the primary beam: (a) 𝜇=0.1, (b) 𝜇=0.2, (c) 𝜇=0.4, and (d) 𝜇=0.8.

It can be seen that the double-beam system successfully suppresses the amplitude at the fundamental frequency of the single beam (at 12 Hz) as a result of countering some of the energy force from the main beam. However, the system now behaves as a two-degree-of-freedom system which, in consequence, creates new resonances. Thus, a significant level of vibration amplitude appears at lower frequency around 8–10 Hz and much lower amplitude level for the second resonance around 25–30 Hz. Increasing the elasticity ratio gives insignificant effect to reduce the amplitude at the first resonance. Instead, this shifts both resonant frequencies to higher frequency, which causes the first resonance to approach that of the single beam; a situation which should be avoided especially if forcing frequency from the primary beam is not stable, for example, a nonsynchronous machine having certain range of operating frequency. Meanwhile, increasing the mass ratio of the double-beam is shown to decrease the frequency of both first and second resonances, although it has less effect on the former. This hence reduces the frequency gap of the two resonances. However, one can observe that the amplitude of the primary beam is reduced with the increasing mass ratio (note the dips just before the second resonance which approaches the fundamental frequency of the single primary beam). It can also be observed that the amplitude at the second resonance increases as the mass ratio is increased.

Figure 3 presents the amplitudes of the primary beam and the absorbing beam. Both amplitudes at the first resonance can be seen not affected by the mass ratio. However, the response of the primary beam at the second resonance is considerably lower than that of the absorbing beam for low mass ratio, but both amplitudes become comparable as the mass ratio increases. As also seen in Figure 2, the compromised situation is that the vibration amplitude of the primary beam becomes lower as the mass ratio increases. The amplitude of the absorber at the second resonance can also be observed to have negligible effect due to the change of mass ratio, and it steeply decreases above this resonant frequency.

(a)

(b)

(c)

(d)

Figure 3 

The vibration amplitude at the midspan location of the double-beam system (𝑒=0.25): (a) 𝜇=0.1, (b) 𝜇=0.2, (c) 𝜇=0.4, and (d) 𝜇=0.8.

4. Effect of Damping and Layer Stiffness

Previous results in Figures 2 and 3 assume very small damping of the visco-elastic layer as well as the structural damping loss factor of the beam. Figure 4 plots the effect of the layer damping, which yields reduction of vibration amplitude only at the second resonance of both the primary beam and the absorbing beam as the damping is increased. At this frequency, the primary beam and the absorber move out of phase for which the role of the visco-elastic layer damping is important to absorb the vibration energy. At the first resonance, as the two beams move in-phase, the layer damping can therefore be seen to have negligible effect on the vibration amplitude.

Figure 4 

Effect of visco-elastic layer damping on the vibration amplitude at the midspan location of the double-beam system (e = 0.25, μ = 0.4, η 1= η 2= 0.01).

Figure 5 shows the effect of the structural damping. It can be seen in Figure 5(a) that increasing the damping of the absorbing beam does not give significant effect to reduce the vibration amplitude. Increasing only the damping of the primary beam reduces only the amplitude at the first resonance, but not at the second resonance as shown in Figure 5(b). The same applies if the damping of the absorbing beam is also increased (see Figure 5(c)). Figure 5(d) shows that the amplitude of the second resonance is mainly controlled by the damping of the visco-elastic layer.

(a)

(b)

(c)

(d)

Figure 5 

Effect of structural damping of the beam on the vibration amplitude at the midspan location of the double-beam system (e = 0.25, μ = 0.4): (a)–(c) ζ = 0.01 and (d) ζ = 0.1.

Figure 6 plots the results for the effect of stiffness of the visco-elastic layer. Here the stiffness is presented using a non-dimensional parameter as a function of the rigidity and length of the main beam, that is, 𝛽=𝑘𝐿⁴/𝐾₁. By reducing the stiffness of the layer, the second resonance shifts to low frequency, while no effect is given to the first resonance. Again same for the case of the layer damping, the in-phase motion of both beams at the first resonance will not be affected by the hardness (stiffness) of the layer. As the mass ratio increases, this frequency shift becomes greater, which also reduces the frequency gap of the two resonances.

(a)

(b)

Figure 6 

Effect of visco-elastic stiffness on the vibration amplitude at the midspan location of the double-beam system (e = 0.25): (a) μ = 0.1 and (b) μ = 0.4.

5. Experimental Validation

Figure 7 shows the schematic diagram and the laboratory setup of the experiment. As in the simulation, the primary beam is placed at the upper side of the absorbing beam. However, it was difficult to realize a simply supported boundary condition. Both beams were therefore clamped to the supporting columns. As the measurement point is at the midspan of the primary beam, the edge condition has negligible effect on the vibration amplitude.

(a)

(b)

Figure 7 

(a) Diagram of the experimental setup and (b) laboratory test.

The primary beam was excited by broadband pseudo-random signal at location close to the midspan of the primary beam using TIRA electromagnetic shaker type TV50018. The input force was measured by a Dytran force gauge type 1051V1. A stud was tightly bolted on the transducer surface and then glued to the beam surface using an epoxy glue. A 200 mm long stinger was used to connect the force gauge and the shaker to minimize the effect of moments transmitted from the shaker. A Dytran accelerometer type 3225F1 was attached exactly at the mid point to measure the vibration amplitude.

The experimental test was conducted with the primary beam made of steel having thickness 20 mm, width 60 mm, and length 2 m. Four measurement cases were made with different configuration and material of the absorber beam as follows:(a)steel single beam without absorber (ST),(b)steel main beam with steel absorber beam; double-identical beams (ST-ST),(c)steel main beam with wood absorber beam (ST-WD), and(d)steel main beam with aluminum absorber beam (ST-AL).A rubber material was used as the viscoelastic layer between the beams having stiffness of 131 kN/m.

The physical parameters of the beam materials are listed in Table 1.

Table 1 

Physical parameters of the materials used in the experiment.

Figure 8 presents the experimental results for the double-beam structure. It can be seen that it demonstrates good agreement with the theory especially around the resonance, although for each case, broader frequency response from the measured results can also be observed, which slightly overestimated the model.

(a)

(b)

(c)

Figure 8 

Comparison of experimental and theoretical results of the vibration amplitude of the double-beam system for the case of (a) ST-ST, (b) ST-WD, and (c) ST-AL.

6. Conclusions

The effect of structural parameters on the dynamic response of a beam structure attached with a beam vibration absorber through a visco-elastic layer under a stationary harmonic load has been studied. The amplitude of the original primary single beam at the fundamental frequency can be considerably reduced. It is found that increasing the rigidity ratio shifts the resulting resonances of the double-system to higher frequency but gives small effect on reducing the vibration amplitude of the resonances. Reducing the mass ratio reduces considerable level of the second resonance of the primary beam as well as widens the gap between the resonant frequencies. However, this increases the first resonance to approach the troublesome resonant frequency of the original single beam but further reduces the vibration of the primary beam, a compromised situation which should be taken into account in the absorber design. The same phenomena apply for the absorbing beam, but it does not affect its vibration amplitude. Adding more damping to the layer has been shown to reduce only the second resonant amplitude. The amplitude at the first resonance can be reduced by increasing the damping of the primary beam. Meanwhile, increasing the layer elasticity (reducing stiffness) reduces the second resonance. The theoretical results have been validated by experimental data with a reasonable agreement.

Appendix

Substituting (3) and (4) into (1) and (2) gives 𝐸₁𝐼₁𝜕⁴𝑋_𝑛(𝑥)𝜕𝑥⁴𝑞_1𝑛(𝑡)+𝑚₁𝑋_𝑛(𝑥)𝑞̈_1𝑛(𝑡)+𝑐𝑋_𝑛(𝑥)𝑞̇_1𝑛(𝑡)−𝑞̇_2𝑛(𝑡)+𝑘𝑋_𝑛(𝑥)𝑞_1𝑛(𝑡)−𝑞_2𝑛(𝑡)=𝑃𝑒^𝑖𝜔𝑡,𝐸₂𝐼₂𝜕⁴𝑋_𝑛(𝑥)𝜕𝑥⁴𝑞_2𝑛(𝑡)+𝑚₂𝑋_𝑛(𝑥)𝑞̈_2𝑛(𝑡)+𝑐𝑋_𝑛(𝑥)𝑞̇_2𝑛(𝑡)−𝑞̇_1𝑛(𝑡)+𝑘𝑋_𝑛(𝑥)𝑞_1𝑛(𝑡)−𝑞_2𝑛(𝑡)=0. (A.1)(A.2)

Substituting (5) into (A.1) and (A.2) with 𝑃=𝐹₀𝛿(𝑥−𝐿/2) yields 𝐸₁𝐼₁𝜎⁴_𝑛𝑋_𝑛(𝑥)𝑞_1𝑛(𝑡)+𝑚₁𝑋_𝑛(𝑥)𝑞̈_1𝑛(𝑡)+𝑐𝑋_𝑛(𝑥)𝑞̇_1𝑛(𝑡)−𝑞̇_2𝑛(𝑡)+𝑘𝑋_𝑛(𝑥)𝑞_1𝑛(𝑡)−𝑞_2𝑛(𝑡)=𝐹₀𝛿𝐿𝑥−2𝑒^𝑖𝜔𝑡,𝐸₂𝐼₂𝜎⁴_𝑛𝑋_𝑛(𝑥)𝑞_2𝑛(𝑡)+𝑚₂𝑋_𝑛(𝑥)𝑞̈_2𝑛(𝑡)+𝑐𝑋_𝑛(𝑥)𝑞̇_1𝑛(𝑡)−𝑞̇_2𝑛(𝑡)+𝑘𝑋_𝑛(𝑥)𝑞_1𝑛(𝑡)−𝑞_2𝑛(𝑡)=0. (A.3)(A.4) Equations (A.3) and (A.4) can be simplified to 𝑚₁𝑞̈_1𝑛(𝑡)+𝑐𝑞̇_1𝑛(𝑡)−𝑐𝑞̇_2𝑛(𝑡)+𝐸₁𝐼₁𝜎⁴_𝑛+𝑘𝑞_1𝑛(𝑡)−𝑘𝑞_2𝑛(𝑡)𝑋_𝑛(𝑥)=𝐹₀𝛿𝐿𝑥−2𝑒^𝑖𝜔𝑡,𝑚₂𝑞̈_2𝑛(𝑡)+𝑐𝑞̇_2𝑛(𝑡)−𝑐𝑞̇_1𝑛(𝑡)+𝐸₂𝐼₂𝜎⁴_𝑛+𝑘𝑞_2𝑛(𝑡)−𝑘𝑞_1𝑛(𝑡)𝑋_𝑛(𝑥)=0. (A.5)(A.6) Multiplying both sides of (A.5) and (A.6) by mode shape function𝑋_𝑚𝑚₁𝑞̈_1𝑛(𝑡)+𝑐𝑞̇_1𝑛(𝑡)−𝑐𝑞̇_2𝑛(𝑡)+𝐸₁𝐼₁𝜎⁴_𝑛+𝑘𝑞_1𝑛(𝑡)−𝑘𝑞_2𝑛(𝑡)𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)=𝐹₀𝛿𝐿𝑥−2𝑋_𝑚(𝑥)𝑒^𝑖𝜔𝑡,𝑚₂𝑞̈_2𝑛(𝑡)+𝑐𝑞̇_2𝑛(𝑡)−𝑐𝑞̇_1𝑛(𝑡)+𝐸₂𝐼₂𝜎⁴_𝑛+𝑘𝑞_2𝑛(𝑡)−𝑘𝑞_1𝑛(𝑡)𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)=0 (A.7)(A.8) Integrating (A.7) and (A.8) through the beam length 𝑚₁𝑞̈_1𝑛(𝑡)+𝑐𝑞̇_1𝑛(𝑡)−𝑐𝑞̇_2𝑛(𝑡)+𝐸₁𝐼₁𝜎⁴_𝑛+𝑘𝑞_1𝑛(𝑡)−𝑘𝑞_2𝑛(𝑡)^𝐿₀𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)=𝐹₀𝑒^{𝑖𝜔𝑡𝐿}₀𝑋_𝑚(𝑥)𝛿𝐿𝑥−2=𝐹₀𝑒^𝑖𝜔𝑡sinn𝜋2𝑚₂𝑞̈_2𝑛(𝑡)+𝑐𝑞̇_2𝑛(𝑡)−𝑐𝑞̇_1𝑛(𝑡)+𝐸₂𝐼₂𝜎⁴_𝑛+𝑘𝑞_2𝑛(𝑡)−𝑘𝑞_1𝑛(𝑡)^𝐿₀𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)=0. (A.9)(A.10)

Applying the orthogonality conditions ^𝐿₀𝑋_𝑛(𝑥)𝑋_𝑚(𝑥)𝑑𝑥=0,𝑛≠𝑚, (A.11) and for n = m, (A-11) gives ^𝐿₀𝑋²_𝑛(𝑥)𝑑𝑥=^𝐿₀sin²𝜎_𝑛𝑥𝑥d𝑥=2−sin2𝜎_𝑛𝑥4𝜎_𝑛^𝐿₂=𝐿2. (A.12)

Equations (A-9) and (A-10) can therefore be expressed in a matrix form in terms of mass ratio (𝜇=𝑚₂/𝑚₁) and rigidity ratio (𝑒=𝐸₁𝐼₁/𝐸₂𝐼₂),𝑚₁00𝜇𝑚₁𝑞̈₁𝑞̈₂+𝑐−𝑐𝑞̇−𝑐𝑐𝑞̇¹₂+𝐾₁𝜎⁴_𝑛+𝑘−𝑘−𝑘𝑒𝐾₁𝜎⁴_𝑛𝑞+𝑘𝑞¹₂=𝐹₀02𝐿sin𝑛𝜋2𝑒^𝑖𝜔𝑡 (A.13)

Nomenclature

𝐸:	Modulus of elasticity of the beam (N/m2)
𝐼:	Area moment of inertia of the beam (m4)
𝑤:	Vertical displacement of the beam (m)
𝑥:	Position coordinate (m)
𝑡:	Time (s)
𝑚:	Mass per unit length (kg/m)
𝑘:	Layer stiffness (N/m2)
𝛽:	Non-dimensional layer stiffness parameter
𝜔:	Radial frequency (rad/s)
𝜔𝑛:	Natural frequency at the nth mode (rad/s)
𝜇:	Mass ratio of absorbing beam to primary beam
𝜁:	Damping ratio of layer
𝑒:	Rigidity ratio of absorbing beam to primary beam
𝐹0:	Magnitude of the external load (N)
𝜎:	Eigenvalue of the mode shape function
𝑋:	Mode shape function
𝑞:	Generalized time function of amplitude (m)
𝑄:	Complex displacement amplitude (m)
𝛿:	Dirac delta function
𝐿:	Length of beam (m)
𝑁:	Newton, unit of force.