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Journal of the Brazilian Society of Mechanical Sciences

Print version ISSN 0100-7386

J. Braz. Soc. Mech. Sci. vol.22 n.3 Campinas  2000

http://dx.doi.org/10.1590/S0100-73862000000300004 

Using Passive Techniques for Vibration Damping in Mechanical Systems

 

Valder Steffen, Jr
Domingos A. Rade

School of Mechanical Engineering. Federal University of Uberlândia. Campus Santa Mônica. 38400-902 Uberlândia. MG. Brazil
Daniel J. Inman
Center for Intelligent Material Systems and Structure. Virginia Polytechnic Institute and State University. Department of Mechanical Engineering. 310 New Engineering Building. MC 0261. Blacksburg. VA 24061. USA

 

 

This paper examines two passive techniques for vibration reduction in mechanical systems: the first one is based on dynamic vibration absorbers (DVAs) and the second uses resonant circuit shunted (RCS) piezoceramics. Genetic algorithms are used to determine the optimal design parameters with respect to performance indexes, which are associated with the dynamical behavior of the system over selected frequency bands. The calculation of the frequency response functions (FRFs) of the composite structure (primary system + DVAs) is performed through a substructure coupling technique. A modal technique is used to determine the frequency response function of the structure containing shunted piezoceramics which are bonded to the primary structure. The use of both techniques simultaneously on the same structure is investigated. The methodology developed is illustrated by numerical applications in which the primary structure is represented by simple Euler-Bernoulli beams. However, the design aspects of vibration control devices presented in this paper can be extended to more complex structures.
Keywords: Passive techniques, shunted piezoelectric, dynamic vibration absorber

 

 

Introduction

The problem of vibration reduction is faced very frequently in a variety of engineering applications and to achieve this goal both active and passive techniques can be used. As passive techniques are considered to be stable, fail-safe and present low power requirement, they were chosen in this paper in opposition to active control systems that require complex amplifiers and associated sensing electronics. Johnson (1995) presents a review of techniques for designed-in passive damping for vibration control. These techniques are based on one of the following damping techniques: viscoelastic materials, viscous fluids, magnetics and passive piezoelectrics. Among the available passive techniques we have decided to use dynamic vibration absorbers attached to the primary structure together with resonant circuit shunted piezoceramics. For the latter technique a piezoceramic material such as lead zirconate titanate (PZT) is bonded to the primary structure and shunted with a passive electric circuit is considered. Both selected techniques present very similar dynamic equations and can be used in passive energy dissipation applications.

A DVA is essentially a mass-spring-damper appendage which, once connected to a vibrating system, is capable of absorbing the vibration energy at a given excitation frequency. The classical theory of dynamic vibration absorbers is presented in the early works of Den Hartog (1934) and Timoshenko et al. (1974) and these devices have been extensively used for attenuating vibrations in different types of machines and structures (Espíndola and Bavastri, 1997; Bavastri et al., 1998).

The possibility of dissipating mechanical energy with piezoelectric material shunted with passive electrical components was investigated for different circuit shunting configurations (Hagood and von Flotow, 1991). The four basic kinds of shunt circuits (inductive, resistive, capacitive and switched) were presented on a recent survey paper by Lesieutre (1998). The inductive shunt or resonant circuit shunt presents a vibration suppression effect that is very similar to the classical vibration absorber. The electrical circuit is designed to dissipate the electrical energy that is converted from mechanical energy by the piezoelectric element.

The parameters of passive devices for vibration suppression (inertia, stiffness and damping for the DVA; inductance and resistance for the RCS) have to be tuned so as to minimize vibrations generated by a single harmonic forcing frequency. It is well known that when passive vibration absorbers are not well tuned they lose efficiency and even slight changes in the parameters can not be allowed. One possible strategy to cope with this problem is to use adaptive DVAs, which have self-tuning capabilities as suggested by Sun et al. (1995). Another strategy is to search for a set of optimal parameters so as to guarantee that the vibration level is minimized over a wide frequency band, provided that design constraints are respected.

In practical applications involving flexible structures with a significant number of modes in the frequency band of interest, it is possible that different techniques have to be used concurrently. In this way it is possible to avoid inherent limitations of these techniques such as actuator voltage limits, free space for shunt installation and feasible parameters values, total mass and space constraints for dynamic vibration absorbers. Tsai and Wang (1999) present a hybrid technique for structural damping using active piezoelectric actuators with passive shunt. In this paper we investigate the use of the technique based in DVAs together with the RCS technique to overcome design difficulties that arise when only one of these techniques is used separately.

This paper is focused on the design of vibration reduction devices using dynamic vibration absorbers and resonant circuit shunts. The parameters of the passive devices for vibration reduction are obtained through optimization procedures based on Genetic Algorithms (GAs). For this purpose the FRFs of the composite structure (primary system + DVA; primary system + shunts or primary system + DVAs + shunts) are used to define performance indexes to be minimized to obtain the optimal system configuration for each case. In the remainder, the basic equations of the two selected approaches are briefly reviewed and the optimization strategy is discussed. Some numerical applications illustrate the main features of the proposed design methodology.

 

Passive Techniques for Vibration Suppression

In this section the basic formulation of two passive techniques for vibration suppression of mechanical systems are briefly presented. Both of them are based on the theory of dynamic vibration absorbers (proof mass dampers), however the first one uses classical substructure coupling methods and the second is based on resonant circuit shunting piezoelectrics to obtain the composite structure.

 

Dynamic Vibration Absorbers and Substructure Coupling Technique

In the context of this paper we consider substructure A as the primary structure, substructure B is seen as the attached DVAs and structure C represents the composite (primary + DVAs) structure. For each configuration, the dynamic flexibility relations are written:

{XA (w)} = [HA (w)]{FA}. (1)

{XB (w)} = [HB (w)]{FB}. (2)

{XC (w)}= [HC (w)]{FC} (3)

where: {XA (w)}, {XB (w)} and {XC (w)} denote the vectors of harmonic displacement response amplitudes of configurations A, B and C, respectively, {FA}, {FB}and {FC} are the vectors of the amplitudes of the harmonic excitation forces and [HA (w)], [HB (w)] and [HC (w)] designate the receptance matrices pertaining to A, B and C, respectively.

Given the FRFs of the two substructures A and B it is possible to determine the FRFs of the assembled structure C, obtained by coupling A and B through a set of coupling coordinates where equilibrium of forces and compatibility of displacements are enforced.

The general characteristics of a single DVA are depicted in Figure 1.

 

 

In the case of simultaneous attachment of any number q of DVAs to different coordinates of the main structure the corresponding FRF matrix is given by:

(4)

where

(5)

In equation (5), the odd lines and columns correspond to the coupled coordinates of substructure B while the even ones correspond to the free coordinates of B.

More details about the technique presented above can be found in Rade and Steffen (1999).

 

Vibration Suppression with Resonant Shunts

If a piezoelectric element is attached to a structure, it is strained as the structure deforms and a portion of the vibration energy is converted into electrical energy. The piezoelectric element behaves electrically as a capacitor and can be combined with a so-called shunt network in order to perform vibration control. Shunting with a resistor and inductor introduces an electrical resonance, which in the optimal case is tuned to structural resonances. This arrangement is shown in Figure 2. The inductor is used to tune the shunt circuit to a given resonant frequency of the structure and the resistor is responsible for peak amplitude reduction of that particular mode. This configuration is very similar to a classical vibration absorber.

 

 

Hagood and von Flotow (1991) give a complete analytical derivation of a resonant shunted piezoelectric in terms of an analogy with a passive dynamic vibration absorber in which a passive second-order electrical system is appended to the dynamics of the undamped structure. Hollkamp (1994) used the aforementioned work and included the influence of viscous damping to obtain the non-dimensional transfer function for the single-mode-damper as given by.

(6)

where is the modal damping; s is the Laplace variable

wE is the natural frequency of the structure (shorted piezoelectric) is the mechanical stiffness of the shunted piezoelectric.

A multi-mode damper can be obtained by introducing an additional shunt for each suppressed mode. Steffen and Inman (1999) developed a methodology for vibration suppression over a frequency band. The shunt parameters are obtained using the FRF of the multi-degree-of-freedom system, which is a combination of the FRFs obtained for each one of the modal coordinates. The coefficients of such a combination are the components of the eigenvectors corresponding to the coordinates where the response is measured. The FRF determined at the m-th coordinate for an excitation force applied at the k-th coordinate is given by

(7)

where [Q]is the modal matrix.

The transfer function for the single-mode-damper represented by equation (6) is written in the Laplace domain. It corresponds to the FRF when transformed to the frequency domain and the equations for the multi-mode damper can be derived taking into account equation (7).

 

Vibration Suppression Using DVAs and Shunts Simultaneously

As mentioned earlier in this paper, shunting with a resistor and inductor, along with the inherent capacitance of the PZT, creates a resonant LRC circuit that is analogous to a classical dynamic vibration absorber. It is interesting to point out that the DVA absorbs kinetic energy and the shunt absorbs strain energy. However, the literature indicates that large shunt inductors are required for typical PZTs (Johnson, 1995) and this characteristic is more evident at lower frequencies, particularly in the vicinity of the first mode (Hollkamp, 1994). One way to overcome the requirement of large inductance is to use synthetic inductors, which require an external power source (Park et al., 1998). Here we propose an alternative by suggesting the use of a hybrid passive technique involving DVAs and shunts simultaneously attached to the same structure. The idea is to use DVAs to suppress vibration in the modes for which the shunt parameters would be unaffordable.

 

Optimization

In this section the general problem of non-linear optimization is defined, the methodology to minimize the objective function is described, and the optimization strategy is presented.

 

Optimization Problem and Genetic Algorithms

We can define the general problem of non-linear optimal design as the determination of the values of design variables xi (i=1,…,n) such that a given objective function f(xi) attains an extreme value while simultaneously all constraints are satisfied (Eschenauer et al., 1997). The classical optimization algorithms are written in such away that the objective function is minimized. However, if an objective function f is to be maximized, one simply substitutes f by –f in the formulation.

The problem above is formulated mathematically as

Min {f({x})/{h({x})} = 0, {g({x})} < 0}, (8)
xeRn

with Rn n-dimensional set of real numbers,

{x} vector of the n design variables,
 f(x) objective function,
{g({x})} vector of p inequality constraints,
{h({x})} vector of the q equality constraints.

The corresponding feasible domain is defined as

X := {{x}ÎRn/{h({X})}= 0, {g({X})} < 0}, (9)

Genetic Algorithms (G.A.) were used in this paper to minimize the objective function because it was observed that classical techniques presented poor convergence in certain cases, due probably to the existence of local minima. Genetic Algorithms are random search techniques based on Darwin’s "survival of the fittest" theories, as presented by Goldberg (1989). Genetic Algorithms have been used to solve difficult problems with objective functions that do not posses properties such as continuity, differentiability, satisfaction of the Lipschitz condition, all over the domain of interest. A basic feature of the method is that an initial population evolves over generations to produce new and hopefully better designs. The elements (or designs) of the initial population are randomly or heuristically generated.

A basic genetic algorithm uses four main operators which are briefly described below (Michalewicz, 1996). The last two operators are called genetic operators:

-Evaluation – the genetic algorithms require information about the fitness of each population member. The fitness measures the adaptation grade of the individual. An individual can be understood as a set of design variables. No gradient or auxiliary information is used, only the value of the fitness function is needed.

-Selection is the operation of choosing members of the current generation to produce the prodigy of the next generation. Then, design which are better as viewed from the fitness function, are more likely to be chosen as parents.

-Crossover is the process in which design information is transferred to the prodigy from the parents. This allows the construction of new individuals from existing ones, enabling new parts of the solution space to be explored. This way, two individuals produce two new individuals.

- Mutation is a low probability random operation used to perturb the design represented by the prodigy. Mutation alters one individual to produce a single new solution that is copied to the next generation of the population to maintain population diversity.

In this paper, the program GAOT – The Genetic Algorithm Optimization Toolbox for Matlab 5 was used. The basic features of this program are presented by Houck et al. (1995).

 

Optimization Strategy

In the previous section it was shown how the FRFs of the assembled system could be expressed in terms of the DVA parameters. It has also been shown how the single-mode damper FRF can be determined as a function of shunt parameters and how to calculate the FRF over a frequency band for the n d.o.f. system which was previously represented by its modal (uncoupled) equations. Once the frequency band of interest and the coordinates at which vibrations are to be attenuated are defined, the FRFs can be used to define performance indexes to be optimized for the selection of optimal DVA and shunt parameter values. It is also possible to impose constraints on either the parameter values or the vibration levels at selected coordinates of the composite structure, or both.

For illustration, let us consider the case where a harmonic force Fk is applied at coordinate k and the response at the coordinate m is to be minimized over a frequency band wL < w < wU, comprising a number of p frequency lines, by attaching a single shunt to the structure. For this situation, some possible performance indexes can be defined as:

J(L, R) = max{abs[Hmk(wm)] (10)

(11)

where wi ,i=1 to p designate weighting factors to be ascribed to each frequency line within the frequency band of interest.

Performance indexes similar to those defined by equations (10) and (11) can be written for the simultaneous optimization of multiple DVAs and multiple shunts. The situation in which DVAs and shunts are to be used in the same structure can also be studied using the above methodology.

In certain cases it is necessary to deal with more general objective functions, which can combine different kinds of performance indexes forming a multi-criterion optimization problem. Multi-criteria optimization reflects the idea in which the optimal design represents the minimization on two or more criteria. The solutions of such an optimization problem are called Pareto optimum solutions. The Pareto optimality concept is stated as: "A vector of x* is Pareto optimal if there exists no feasible vector x which would decrease some objective function without causing a simultaneous increase in at least one objective function". This means that problems with multiple objective functions are characterized by the existence of conflict, i.e., none of the possible solutions allows for the optimal fulfillment of all objectives for the same design configuration.

To describe mathematically the multi-criteria optimization problem, f(x) given in equation (8) represents now a vector of r objective functions such that

f(x) = (fl (x) ... fr (x))T (12)

There are different ways to transform the vector of objective functions given by equation (12) into a single scalar function to be minimized (Ozyczka, 1990). One possible strategy, which has been used in some numerical applications that will be presented later in this paper, is known as Global Criterion Method. According to this method, all the individual performance indexes have to be first minimized separately to obtain a set of solutions fi0, i = 1 to r. The global criterion to be minimized in this case is given by

(13)

Different values of b can be tested and the one that leads to the most satisfactory solution is selected. The value b = 2 was successfully used by the authors in various applications.

 

Numerical Applications

To illustrate the main features of the methodology proposed in this paper, three numerical applications are presented in this section. The first example is related to the design of two DVAs to be installed on a free-free beam in order to reduce vibrations in two selected frequency bands; the second example is devoted to the situation in which three shunted PZTs are installed on the structure to reduce the vibrations corresponding to the first three modes of a simply-supported beam and the third example shows the possibility of using one DVA and two shunted PZTs on the same structure.

 

Application 1 (Rade and Steffen, 1999)

The FRFs of the primary system were obtained experimentally using standard modal analysis techniques performed on a simple steel uniform free-free beam as shown in Figure 3. For acquisition of the FRFs the excitation was introduced in the vertical direction with an impact hammer and the time domain acceleration responses were collected using piezoelectric accelerometers. The complete set of FRFs relating the four measurement stations whose positions are shown in Figure 3 was acquired within the band [0-2000 Hz], with a frequency resolution of 2.5 Hz. The main characteristics of the instrumentation used are given as follows:

- piezoelectric accelerometer B&K model 4375, sensitivity 0.344 pC/(ms-2)
- piezoelectric force transducer B&K model 8200, sensitivity 3.82 pC/N
- impact hammer B&K model 8202
- charge amplifier B&K model 2635
- two channel spectrum analyser Scientific Atlanta SD380.

 

 

Two DVAs are to be simultaneously attached at coordinate number 1 for absorbing the vibrations due to an excitation force applied at this same coordinate within two disconnected frequency bands. The main aspects of the optimization procedure are:

- frequency bands: [520-680 Hz] and [880-1060 Hz], comprising the third and fourth modes, respectively;
- constraints on absorber masses: total absorber mass less than 10% of the mass of the beam;

- initial parameter values: 

m10 = 0.10kg, c10 = 10Ns / m,k10 = 1.0x106 N / m;
m20 = 0.10kg, c20 = 10Ns / m,k20 = 2.0x106 N / m.

- performance index:

- optimal parameter values:

m1 = 0.124kg; c1 = 111.28Ns / m,k1 = 1.5x106 N / m;
m2 = 0.125kg; c2 = 25.73Ns / m,k2 = 4.4x106 N / m.

Figure 4 shows the FRF H11(w ) for the primary structure alone and for the composite structure (primary + DVAs). Vertical lines bounds the two frequency bands used in the optimization procedure. As expected, the resonant amplitudes of the third and fourth vibration modes were eliminated from the FRF.

 

 

Application 2 (Steffen and Inman, 1999)

A simply supported beam with the following properties is considered: Eb=70 GPa; rb = 2500KG / m3; Lb=0.5m; tb=0.01m and b=0.05m. We consider the case in which three shunted PZTs are installed on the structure at the position corresponding to the maximum strain energy for each mode, respectively. The inherent capacitance of each PZT is taken as Cp=150 nF. The generalized coupling coefficient is considered to be K=0.12. For the determination of the FRF, it was considered that the harmonic force was applied at node number 2 and the system response was determined at this same node. Table 1 shows the optimal parameters obtained for the shunts using genetic algorithms. Figure 5 shows the numerical results of the frequency response functions with and without the optimal shunts.

 

 

 

Application 3

Application we use the same dynamical system as described in the previous application. However, instead of using three shunts to reduce the vibration of the first three vibration modes, a hybrid passive technique is proposed. This way, a DVA will be used to reduce the vibrations associated to the first mode and two shunted PZTs will reduce the remaining two modes of interest. Table 2 shows the optimal parameters obtained using genetic algorithms.

 

 

Figure 6 shows the system FRF in the frequency band encompassing the first natural frequency. A DVA was used to reduce the vibrations in this frequency band. Figure 7 shows the system FRF in the frequency band encompassing the second and third natural frequencies. As explained earlier in this paper, two shunted PZTs were installed to reduce the vibrations in this frequency band.

 

 

 

It can be observed from this procedure that using one DVA and two shunted PZTs all the resulting optimal parameter values for the two types of vibration absorbers used are feasible, and the use of a large 18.2H inductor is no longer required.

 

Conclusions

This paper presented a general methodology for the optimal design of passive devices for vibration reduction in mechanical systems when several natural frequencies are present in the frequency band of interest. Two different passive techniques were explored: dynamic vibration absorbers and shunted PZTs. A classical substructure coupling method is used to determine the FRF of the resulting composite structure (primary + DVAs). Using a multi-mode damper technique, which includes an additional shunt for each suppressed mode, it is possible to obtain the FRF of the n d.o.f. system based on a modal formulation. To avoid unaffordable shunt parameters, this paper presents the possibility of using DAVs and RCSs simultaneously attached to the same structure, creating a hybrid passive technique for vibration suppression. The general non-linear optimization problem was formulated and genetic algorithms were used to minimize single or multi-objective functions, which are formulated using FRF relations. The applications showed three possible issues for vibration reduction in mechanical systems: using multiple DVAs attached to the primary structure, using multiple shunts installed at maximum strain positions and using DVAs and RCSs simultaneously on the same structure. This way it is possible to avoid inherent limitations of the presented passive techniques. Frequently found limitations are free space for shunt installation, feasible parameter values, total mass and space constraints for dynamic vibration absorbers, etc. The results presented are encouraging, the techniques are effective over a frequency band of interest and the methodology developed can be extended to more complex structures. Future work on this topic will focus on a unified approach for the design of passive vibration reduction techniques, including dynamic vibration absorbers and shunted PZTs.

 

Acknowledgements

Dr. Steffen is thankful to Capes Foundation (Brazil) and to Fulbright Program (USA) for the scholarship awarded during his visit at Virginia Polytechnic Institute and State University in 1999. Dr. Rade acknowledges the Brazilian agency CNPq for the financial support to his research through the award of a research scholarship. Dr. Inman gratefully acknowledges the support of National Science Foundation award number CMS-9713453-001 and Air Force Office of Scientific Research grant number F49620-99-1-0231.

 

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Manuscript received: May 1999. Technical Editor: Paulo Eigi Miyagi.

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