Vibration Isolation of Lumped Masses Supported on Beam by Imposing Nodes Using Multiple Vibration Absorbers

In this paper, variable stiffness damped absorbers are used to isolate the substructures of Euler-Bernoulli beam, modelled as lumped masses, from vibrations. The novel algorithm is developed that can be used to determine the required absorber masses and resonance frequencies to impose nodes at selected locations on beam with the constraint of vibration amplitude of absorber mass. Numerical simulations are performed to show the effectiveness of the proposed algorithm. Experimental test is conducted on a cantilever beam with two absorbers to verify the numerical results.


Introduction
The tuned vibration absorber (TVA) was invented by (Frahm, 1911), since then, has been is an important engineering tool for vibration suppression.The first classical paper on dynamic vibration absorber was by (Ormondroyd & Den, 1928).Their vibration absorber consisted of a tuned spring-mass used to suppress the response of a harmonic oscillator.An excellent survey of passive, semi-active and active dynamic vibration absorbers was prepared by (Sun et al., 1995).Spring-mass systems which are used as vibration absorbers to minimize excesses vibrations in continuous structure have received considerable interest in recent years.(Young, 1928) was the first to consider the application of absorber to control the vibrations of a continuous structure i.e. cantilever beam, at the absorber attachment point with the absorber tuned to the first natural frequency of the beam.(Manikanahally & Crocker, 1993) employed vibration absorbers to suppress any number of significant modes.The method was successfully applied to a space structure modeled as a mass-loaded free-free beam subjected to localized harmonic excitation.(Esmailzadeh & Jalili, 1998) presented a procedure in designing DVA for a structurally damped beam system subjected to distributed force excitation.The absorber is modeled as spring-mass-damper system and the optimum tuning and damping ratios are determine to minimize the beam dynamic response at the resonance frequency at which they operate.Research on suppressing vibration in a region or the particular span of an elastic structure by using the spring mass vibration absorber has been reported recently.(Cha, 2004(Cha, , 2005) ) employed spring-mass vibration absorber to reduce vibrations at desired locations by imposing node technique (Hao et al., 2011).Suppressed the hand-arm vibration in electric grass trimmer by installing tunable vibration absorber at optimum location.The focus of (Cha & Rinker, 2012) was on enforcing nodes at desired location of damped Euler-Bernoulli beam during forced harmonic excitations using damped vibration absorbers.An efficient method is developed which determines the restoring force exerted by the damped absorbers using Gaussian elimination and then the forces are used to determine the parameters of oscillators.(Hao & Ripin, 2013) applied imposing node technique to achieve very low vibration at handle location using two tunable vibration absorbers.The grass trimmer system is simplified as an arbitrary, supported beam constrained by 4 lumped masses.The Matlab routine global search is utilized to obtain the tuning frequencies of the absorbers.(Patil, & Awasare, 2016) developed an iterative procedure to find the required resonance frequencies of absorbers to impose node at selected locations on beam.
In the method proposed by Cha to induce multiple nodes, a set of nonlinear algebraic equations need to be solved simultaneously.Numerically, the solution of these equations is very computationally intensive because the convergence is often very slow.The limitations of the procedure developed by Patil & Awasare are that, the maximum allowable absorber amplitudes were not considered while finding the resonance frequencies of the absorbers.The purpose of this paper is 1) To develop mathematical model of the beam carrying multiple damped absorbers and to formulate the equation for imposing nodes at desired locations on the beam 2) To develop an algorithm to find the required absorber parameters to impose nodes at selected locations on the beam with constraint of the tolerable vibration amplitudes of the absorber mass.3)To perform numerical simulation on beam to show the utility of the proposed algorithm 4) To carry the experimental test to validate the numerical results.

Theory
Figure 1 shows an arbitrarily supported, Euler-Bernoulli beam with n tunable vibration absorbers attached at i x .
The absorber modeled as single degree of freedom spring-mass-damper system having mass i m , stiffness i where N is the number of modes used in the assumed-modes expansion, ( ) i x φ are the eigenfunctions of the undamped beam and ( ) i t η are corresponding generalized co-ordinates.Applying Lagrange's equations and assuming simple harmonic motion with same response frequency as the excitation frequency, the following equations of motion are obtained , and the matrices [m] [c] and [k] of size n n × are diagonal, whose i th elements are given by i m , i c and i k respectively.The N N Using second equation of equation ( 2), the i z are found to be 2 2 2 ( ) , In equation ( 6) resonance frequency i ω and damping coefficient i c of i th absorber are given by where i ζ is the damping ratio of the i th absorber Equation ( 6) is substituted into the first equation of equation ( 2) and then solving for η to obtain In equation ( 8) Substituting equation ( 8) in to equation (1), following equation is formulated, the solution of which gives the absorber parameters required to impose node at desired locations nr x along the beam Once the beam with its boundary conditions are specified, absorbers attachment locations , i.e the absolute value of the displacement of the beam at node location is less than the absolute value of the displacement at previous and next to node location as shown in Figure 2. The condition ( ) ( ) gives the absorber frequency ω for given mass m necessary to impose node.
The procedure to find the absorber masses 1 m and 2 m and frequencies of the absorbers 1 ω and 2 ω to impose two nodes is as follows Algorithm to find the masses and corresponding resonance frequencies of the absorbers to impose two nodes (1) Assume the lower value for the absorber masses 1 m and 2 m .
(7) Increase the mass 1 m of first absorber and repeat steps (3) to ( 5 , till above condition is achieved.

Numerical Results
Because the assumed-mode method was used to formulate the equations of motions, the proposed procedure can be easily implemented to impose node along any arbitrary supported beam subjected to harmonic excitations.For cantilever beam, its normalized (with respect to mass per unit length, ρ , of the beam) eigenfunctions ( ) i x φ , generalised masses i M and generalised stiffnesses i K are given by sin sinh 1 ( ) cos cosh (sin sinh ) cos cosh where i L β satisfies the following transcendental equation cos cosh ) where E is Young's modulus, I is the moment of inertia of the cross-section of the beam.
In the following example the frequencies and and vibration amplitudes are non-dimensionalised by dividing by on the beam.The absorber parameters, resonance frequencies and masses for given damping ratio, required to impose nodes obtained by using the algorithm developed are listed in Table -1.Table 1.Summary of absorber parameters for the example of vibration isolation of lumped masses supported on uniform cantilever beam Absorber Damping ratio L Note the region on beam up to 0.6L experiences less vibration comparing to the beam without absorbers.
Figure 3.The absolute steady-state response of beam with and without absorbers to impose nodes at 0.4L and 0.6L , for the absorber parameters listed in Table 1

Experimental Test
For imposing two nodes the dual cantilevered mass absorbers were designed and constructed as shown in Figure 5.The resonance frequency of device is adjusted by moving the masses towards or away from the base support, which alters the effective stiffness in the system and alters its resonance frequency.Note that the structural damping of the absorbers used is considered as equivalent viscous damping.Next in order to verify the numerical result, experimental test was conducted for the case of cantilever beam with two absorbers as shown in Figure 5.The two absorbers are attached at 0.3L and 0.65L and the harmonic input force is applied at the tip of the beam with frequency 152 Hz i.e F EI L .The system parameters and material properties used in experimental test are listed in Table 2.The toltal weights of absorber end masses attached at 0.3L and 0.65L are 0.25 Kg and 0.42 Kg respectively.The damping ratios of the both absorber are 1 2 0.001 . The absorbers were tuned by moving the end masses in or out such that the displacement at 0.4L and 0.6L was minimized.After tuning of the absorbers the vibration amplitudes were measured at twenty points on the beam's surface by the accelerometer and recorded by vibration analyzer to plot experimental steady state response as shown in Figure 6.It is observed that the vibrations at the node locations of the beam are reduced to a minimum level.Comparing Figure 3, with Figure 6, it is observed that there is good agreement between the numerical results and experimental results.

Conclusions
This investigation presents novel algorithm to find the absorber parameters to impose nodes at chosen locations on beam.Once the generalized program is developed, it can be used to determine the feasible absorber parameters and can be easily modified to accommodate beam with different boundary conditions.For imposing multiple nodes, the procedure gives different combinations of two absorber masses and corresponding resonance frequencies required to impose nodes for given damping ratio.The design constraint on maximum allowable vibration amplitudes on absorber masses makes the proposed procedure more practical.The results show that by imposing nodes at appropriate locations, the vibrations are suppressed for the segment of beam thus isolating the lumped masses.The experimental results show good agreement with those obtained by numerical experiments.

Figure 1 .
Figure 1.Beam with variable stiffness damped multiple absorbers subjected to localized harmonic excitations 1

K
are diagonal matrices whose i th elements are , i M , i C and i K are the generalized masses, damping and stifnesses of beam.Vector of the eigenfunctions of the beam and the matrices [ ]

ω
and the excitation location f x are known, equation (10) can be used to find the absorber parameters, mass of the absorber i m and resonance frequencies i ω , for given absorber damping ratio i ζ at which displacements of beam ( ) nr W x becomes zero to impose nodes at nrx .An algorithm is developed, which is based on finding the resonance frequency of absorber ω at which ( )

( 12 )
Increase the mass 2 m of second absorber and repeat steps (8) to (10) till, Replace frequency and mass of second absorber in 2 σ with new frequency 2 ω and mass 2 m obtain from step number (11).(14) Repeat procedure from step number (3) to (12) with revised 2 σ .

Figure
Figure 2. Illustration of condition

F
EI L respectively.Number of modes, 15 N = is used in the assumed-modes expansion.The value of the absorber frequency i ω and mass i m are 0001 L ρ , in each iteration respectively.The absorber has a low damping to obtain the greatest vibration attenuation at the intended frequency.Now consider the example of a uniform cantilever beam.It is desired that two nodes to be imposed, at 1

Figure 3
Figure3, shows the steady state deformed shapes of cantilever beam with node at 0.4L and 0.6 .L Note the region on beam up to 0.6L experiences less vibration comparing to the beam without absorbers.

.
The force amplitude is kept constant at 5 N through the experiment and used to non-dimensionalize the displacements of the beam by dividing