JP6498924B2 - Double dynamic vibration absorber and design method of double dynamic vibration absorber - Google Patents

Description

The present invention relates to a double dynamic vibration absorber and a method for designing a double dynamic vibration absorber.

  Machines are becoming smaller and lighter, resulting in lower rigidity and more vibration. Therefore, the importance of vibration control technology is increasing. In addition, as a vibration damping technique that can be reduced in size and weight, there is a vibration damping technique using a bending vibration type dynamic vibration absorber.

JP 2006-207749 A

Keisuke Yamada, Hiroshi Matsuhisa, Hideo Utsuno, and Katsutoshi Sawada, “Optimum tuning of series and parallel LR circuits for passivevibration suppression using piezoelectric elements”, Journalof Sound and Vibration, Vol. 329, No. 2 (2010), pp. 5036-5057 . Keisuke Yamada, Hiroshi Matsuhisa, Hideo Utsuno, “Double Dynamic Absorber Using Piezoelectric Element and Electric Circuit”, Transactions of the Japan Society of Mechanical Engineers (C), 73, 730 (2007), pp. 1633-1640. J. Ormondroyd and J. P. Den Hartog, “The theory of the dynamicvibration absorber”, Transactions of the American Society of Mechanical Engineers, Vol. 50, No. 7 (1928), pp. 9-22. Keisuke Yamada, “Analysis by replacing displacement excitation with equivalent force excitation at boundary of continuum”, Dynamics and Design Conference (2014). Matsuhisa Hiroshi, Zhang Imeaki, Tajima Junya, Honda Yoshihisa, Sato Susumu, “Cantilevered Dynamic Absorber with Force and Moment Working”, Transactions of the Japan Society of Mechanical Engineers (C), 54, 508 (1988), pp 2935-2941. Taichi Matsuoka, Tetsuya Takabe, Yujiro Tsuji, Nobuhiro Wakabayashi, Ichiro Otsuki, “Suppression of beam vibration by moment-type dynamic vibration absorber”, Transactions of the Japan Society of Mechanical Engineers (C), 78, 792 (2012), pp. 2733-2745.

  However, it is difficult to add damping to the vibration control method using the bending vibration type dynamic vibration absorber. In this regard, Non-Patent Document 5 states that it is necessary to use a magnetic damper to control attenuation. In addition, when a magnetic damper is used, there exists a problem that a dynamic vibration absorber will enlarge. In Non-Patent Document 6, damping is added using an acrylic double-sided adhesive tape sandwiched between two copper plates as a viscoelastic body, but with a special structure (a structure in which a viscoelastic body is sandwiched). There is a problem that a bending vibrator is required and is difficult to put into practical use.

  On the other hand, there is a vibration damping device disclosed in Patent Document 1 as an example of a vibration damping device that does not use a magnetic damper or a viscoelastic body. However, the vibration damping device disclosed in Patent Document 1 is provided with a piezoelectric element in a mass damper and actively applies a control voltage to the piezoelectric element to suppress vibration of a vibration damping object. It is not a configuration for adding attenuation. Further, the vibration damping device disclosed in Patent Literature 1 requires a power source for driving a controller that generates a control voltage, and there is a problem that the system may become unstable.

In view of the above situation, an object of the present invention is to provide a double dynamic vibration absorber capable of functioning as a bending vibration body type dynamic vibration absorber with damping and a design method thereof.

  In order to achieve the above object, a dynamic vibration absorber according to the present invention is a dynamic vibration absorber that is attached to a vibration body that is a vibration suppression target and suppresses vibration of the vibration body, and has one end attached to the vibration body. A bending vibration body, a piezoelectric element that is attached to the bending vibration body and converts a load applied to the bending vibration body into a voltage, and an electric circuit connected between electrodes of the piezoelectric element, the electric circuit comprising: A configuration (first configuration) including an inductor and a resistor is adopted.

  In order to achieve the above object, a dynamic vibration absorber design method according to the present invention is a dynamic vibration absorber design method according to the first configuration, in which attenuation by an electric system including the piezoelectric element and the electric circuit is performed. Performing an approximation to replace the effect with a linear dashpot; and deriving a governing equation when the approximation of the dynamic vibration absorber is performed using an equivalent mechanical model when the approximation of the dynamic vibration absorber is performed And using the evaluation index obtained from the governing equation when the approximation is performed, analytically obtaining an optimal adjustment formula for the dynamic vibration absorber, a reference value obtained from the optimal adjustment formula, or from the reference value And determining a part of the specifications of the dynamic vibration absorber using an adjustment value obtained by adjusting the value (second configuration).

  Further, in the design method of the second configuration, the step of obtaining the optimum adjustment formula may be performed when the fixed vibration theory is applied using an evaluation index obtained from a governing equation when the approximation is performed. Substeps for obtaining the optimum values of the spring constant and damping constant, and the analytical model in which the dashpot is returned to the original electrical system and the mass point of the vibrating body is fixed in space, is obtained from the governing equation of the analytical model. A sub-step of obtaining respective optimum values of the inductance of the inductor and the resistance value of the resistor when the fixed point theory is applied using an evaluation index may be adopted (third configuration).

  In the design method of the second or third configuration, the evaluation index may be any one of compliance, mobility, and acceleration (fourth configuration).

  In order to achieve the above object, another design method of the dynamic vibration absorber according to the present invention includes a step of performing an approximation to replace a damping effect by an electric system including the piezoelectric element and the electric circuit with a linear dashpot; Deriving a governing equation when the approximation of the dynamic vibration absorber is performed using an equivalent mechanical model when the approximation of the dynamic vibration absorber is performed, and an evaluation obtained from the governing equation when performing the approximation Analyzing an optimal adjustment formula of the dynamic vibration absorber analytically using an index, and numerical optimization using an evaluation index determined from a governing equation when the approximation of the dynamic vibration absorber is not performed as a predetermined condition Using the step of correcting the optimum adjustment formula, the reference value obtained from the optimum adjustment formula corrected by the numerical optimization, or the adjustment value obtained by adjusting the value from the reference value Determining a portion of the specifications of kidou vibration absorber, a structure comprising a (fifth configuration).

  In the design method of the fifth configuration, the evaluation index is any one of compliance, mobility, and acceleration, and the predetermined condition is a condition that minimizes the maximum value of the evaluation index (first 6 configuration).

  Moreover, in the dynamic vibration absorber of the first configuration, it is preferable to adopt a configuration (seventh configuration) in which a part of the specifications is determined by the design method of any one of the second to sixth configurations. .

ADVANTAGE OF THE INVENTION According to this invention, it becomes possible to provide the double dynamic vibration absorber which can be functioned as a bending-vibration body type dynamic vibration absorber with attenuation | damping, and its design method.

The figure which shows the analysis model of the dynamic vibration damper which concerns on one Embodiment of this indication The figure which shows the analysis model of the state which short-circuited between the electrodes of the piezoelectric element 4 in the dynamic vibration damper which concerns on one Embodiment of this indication. The figure which shows the equivalent mechanical model using the force transducer of the model shown to FIG. 1B The figure which shows the equivalent machine model which does not use the force transducer of the model shown to FIG. 1B The figure which shows the equivalent mechanical model using the force transducer of the model shown to FIG. 1A Diagram showing a simplified equivalent machine model of the model shown in FIG. 1A Diagram showing equivalent machine model using dashpot instead of electric system 1 is an equivalent machine model when the mass point of the main system is fixed in a space, and shows an equivalent machine model of the model shown in FIG. 1A This is an equivalent machine model when the mass point of the main system is fixed in space, and shows an equivalent machine model using a dashpot instead of an electric system The figure which shows the result of the simulation using the analytical optimal adjustment type | formula of the dimensionless compliance magnitude | size of the dynamic vibration absorber shown to FIG. 1A The figure which shows the result of the simulation using the numerical optimal adjustment type | formula of the dimensionless compliance magnitude | size of the dynamic vibration damper shown to FIG. 1A Diagram showing the outline of the experimental apparatus Diagram showing simulation results of dimensionless compliance size Diagram showing experimental results of dimensionless compliance size The figure which shows the modification of the dynamic vibration damper which concerns on one Embodiment of this indication The figure which shows the other modification of the dynamic vibration damper which concerns on one Embodiment of this indication

Hereinafter, preferred embodiments of the present disclosure will be described in detail with reference to the appended drawings. The description will be made in the following order.
1 Theoretical analysis of a dynamic vibration absorber according to an embodiment of the present disclosure
1.1 Derivation of governing equations
1.2 Equivalent machine model
1.3 Compliance
1.4 Optimal adjustment of double dynamic vibration absorber using fixed point theory
1.5 Numerical optimization of double dynamic vibration absorber
1.6 Verification of optimal adjustment formula by simulation
2 Verification by experiment and simulation of a dynamic vibration absorber according to an embodiment of the present disclosure
3 Summary of dynamic vibration absorber according to an embodiment of the present disclosure
4 Modifications

<< 1 Theoretical analysis of a dynamic vibration absorber according to an embodiment of the present disclosure >>
An analysis model of a dynamic vibration absorber 1 (hereinafter, abbreviated as “dynamic vibration absorber 1”) according to an embodiment of the present disclosure is shown in FIG. 1A. In this embodiment the cantilever beam of length l _b of the mass point is placed on the free end of the mass of the main system 2 is damped of the dynamic vibration reducer 1 M. The dynamic vibration absorber 1 includes a cantilever 3 having a length l _d in which a mass point of mass m is installed at a free end, a piezoelectric element 4, an inductor 5, and a resistor 6.

In the present embodiment, the fundamental vibration mode of bending of the main system 2 is damped using the fundamental vibration mode of the dynamic vibration absorber 1. In the present embodiment, the fundamental vibration mode of the main system 2 is damped. However, any order mode of the main system 2 can be damped using the fundamental vibration mode of the dynamic vibration absorber 1 in the same way. Further, in the present embodiment, the secondary and subsequent vibration modes of the main system 2 and the dynamic vibration absorber 1 are approximately ignored. For the _xb axis, the fixed end of the main system 2 is the origin, and the right direction in the axial direction is positive. Similarly, for the _xd axis, the installation point of the dynamic vibration absorber 1 is the origin, and the axial direction of the dynamic vibration absorber 1 is positive. Further, an external force f is applied to the main system 2 at a position of x _b = x _f . The eigenfunction of the fundamental vibration mode of the main system 2 is Ψ _b (x _b ), the mode displacement is ξ _b , the eigenfunction of the fundamental vibration mode of the dynamic vibration absorber 1 is Ψ _d (x _d ), and the mode displacement is ξ _d . The length, width, and thickness of the piezoelectric element 4 are assumed to be l _p , w _p , and t _p , respectively. The x _d coordinates of the left and right ends of the piezoelectric element 4 are assumed to be x _L and x _R (= x _L + l _p ), respectively. In the present embodiment, it is assumed that the main system 2 has a small attenuation and the dynamic vibration absorber 1 needs to suppress the vibration, and the main system 2 is ignored.

  In the present embodiment, a plate-type piezoelectric element is used as the piezoelectric element 4, but there is no essential difference even if the configuration is other than this including the type of the piezoelectric element. Further, in the present embodiment, the case where the dynamic vibration absorber 1 is installed at the end of the main system 2 is considered, but the same can be considered when it is installed at a position other than the end. The main system 2 is not limited to a cantilever and may be a vibrating body. In the present embodiment, the cantilever beam 3 is used as an example of the bending vibrator, but it may have a shape other than the beam shape, and any boundary condition may be used for the boundary condition.

  In this embodiment, first, the equation of motion in a state where the electrodes of the piezoelectric element 4 shown in FIG. 1B are short-circuited is derived, and the LR series circuit including the inductor 5 and the resistor 6 is connected to the piezoelectric element 4 using the equation of motion. Find the equation of motion. Although the analysis model is a three-degree-of-freedom vibration system, there is no known technique that analytically derives the optimum adjustment formula for a double dynamic vibration absorber, and all of them use numerical optimization to obtain the optimum value (for example, non-patent Reference 2). Also in this embodiment, the optimum adjustment formula is finally obtained by a numerical optimization method. However, for practical convenience and understanding of the dynamic vibration absorber 1, the adjustment formula is obtained analytically using approximation. . In this embodiment, the adjustment formula to which the analytical optimization method is applied is referred to as an analytical optimal adjustment formula, and is distinguished from the numerical optimum adjustment formula to which the numerical optimization method is applied. The effectiveness of the analytical optimum adjustment formula and numerical optimum adjustment formula is verified by simulation.

<1.1 Derivation of governing equations>
When the electrodes of the piezoelectric element 4 are short-circuited as shown in FIG. 1B, only mechanical rigidity works for the piezoelectric element 4. In this embodiment, the equation of motion is obtained using the Lagrange method under these conditions, and the equation of motion when the LR series circuit is connected to the piezoelectric element 4 is obtained using the result. When the electrodes of the piezoelectric element 4 are short-circuited as shown in FIG. 1B, the kinetic energy _TE is expressed by the following equation (1).
Here, ρ _b is the density of the beam portion of the main system 2, w _b is the width of the beam portion of the main system 2, t _b is the thickness of the beam portion of the main system 2, and ρ _d is the cantilever beam 3 , W _d is the width of the cantilever 3, t _d is the thickness of the cantilever 3, and the mass of the piezoelectric element 4 is neglected as being sufficiently smaller than the mass of the cantilever 3.

Further, the potential energy U _E of the analytical model is expressed by the following equation (2).
Here, E _b is the Young's modulus of main system 2, I _b is the secondary moment of inertia of main system 2, E _d is the Young's modulus of cantilever beam 3, I _d is the secondary moment of inertia of cantilever beam 3, k _p is a mechanical spring constant in the longitudinal direction of the piezoelectric element 4, and θ _k is an influence coefficient of the piezoelectric element 4.

The main system 2 second moment I _b, second moment I _d of 3 Hari cantilever, the piezoelectric element 4 longitudinal mechanical spring constant k _p, of the piezoelectric elements 4 influence coefficient theta _k is the following, respectively Equations (3)-(6) are obtained.
Where E _p is the Young's modulus of the piezoelectric element 4. Here, for the sake of simplicity, the change of the neutral axis of the beam due to the attachment of the piezoelectric element 4 is ignored, and the neutral axis coincides with the center line in the thickness direction.

Further, the virtual work δW _f due to the external force is expressed by the following equations (7) and (8).
Here, δξ _b is a virtual mode displacement of the mass point of the main system 2.

From equations (1), (2), and (7), Lagrange's equation of motion becomes the following equations (9)-(16).

Here, when the equations of motion (9) and (10) are arranged in the following equations (17) to (19), the following equations (20) and (21) are obtained.

<1.2 Equivalent machine model>
When the following equations (22)-(27) are introduced from the equations of motion (20), (21), the equations of motion become the following equations (28), (29).

Here, a model in which the electric circuit shown in FIG. 1B is short-circuited from the equations (20) and (21) is expressed as an equivalent mechanical model using a force transducer as shown in FIG. 2A. The equations of motion (28) and (29) indicate that they are equivalent to the equivalent mechanical model of FIG. 2B. Θ _b given by Equation (19) is an influence coefficient of the dynamic vibration absorber 1 on the main system 2.

The equivalent mechanical model of FIG. 2A is a case where the electrodes of the piezoelectric element 4 are short-circuited, but when an LR series circuit is connected as shown in FIG. 1A, one degree of freedom in parallel with the mechanical spring of the piezoelectric element 4. This is a model in which a vibration system is installed (see Non-Patent Document 1). Therefore, FIG. 3A shows an equivalent mechanical model when a double dynamic vibration absorber (dynamic vibration absorber 1) using the piezoelectric element 4 is installed. Here, L is the inductance of the inductor 5, R is the resistance value of the resistor 6, C _p ^S is the capacitance of the piezoelectric element 4 alone under a constant strain, θ _p is the electromechanical coupling coefficient of the piezoelectric element 4 alone, k _e is the electrical spring constant of the piezoelectric element 4, the piezoelectric constant of the piezoelectric elements 4 and d _31, electrical spring constant k _e of the piezoelectric element 4 single electromechanical coupling coefficient theta _p and the piezoelectric element 4 Are the following equations (30) and (31) (see Non-Patent Document 1).

From the equivalent machine model of FIG. 3A, the governing equation is obtained as the following equations (32)-(37).

When governing equations (32)-(34) are arranged, the following equations (38)-(43) are obtained.
Expressions (38)-(40) are represented by an equivalent machine model as shown in FIG. 3B.

<1.3 Compliance>
The dimensionless compliance from the governing equations (38)-(40) obtained in the previous section is given by the following equations (44)-(54). It should be noted that, when the optimal value is derived analytically, if dimensionless is arranged, it is organized and easy to solve. Therefore, in this embodiment, dimensionless compliance is used, but dimensionless is not essential in the evaluation index.
Here, F is the amplitude of the external force f _ex and Ξ ₁ is the complex amplitude of the mode displacement ξ ₁ .

<1.4 Optimum adjustment of double dynamic vibration absorber using fixed point theory>
In this section, an approximate optimal adjustment formula for a double dynamic vibration absorber (dynamic vibration absorber 1) is obtained using approximation. It can be considered that the electric system composed of the piezoelectric element 4 and the LR series circuit adds attenuation to the first dynamic vibration absorber (cantilever 3). Therefore, this is replaced with a linear dashpot as shown in FIG. 4 to obtain the optimum spring constant and damping coefficient of the dynamic vibration absorber. Since FIG. 4 is the same as the analysis model of a conventional single-motion vibration absorber, an optimum value can be obtained using a fixed point theory (see Non-Patent Document 3).

Next, when the replaced dashpot is returned to the original electric system and the mass point of the main system 2 is fixed in the space, a two-degree-of-freedom vibration system is obtained as shown in FIG. 5A. This uses a piezoelectric element 4 and an LR series circuit. Since it is the same as the analysis model of passive vibration suppression, the optimum values of inductance and resistance can be obtained using fixed point theory (see Non-Patent Document 1). Further, the optimum value of the equivalent stiffness ratio β is obtained using a condition in which the magnitude of damping is the same in the case of FIG. 5A and the case where the electric system is replaced with a dashpot as shown in FIG. 5B. Specifically, when an external force acts on the mass point of the mass M ₂ in FIGS. 5A and 5B, the optimum value of the equivalent stiffness ratio β is obtained under the condition that the maximum amplitudes of both are equal.

When the electric system composed of the piezoelectric element 4 and the LR series circuit is replaced with a dash pot as shown in FIG. 4, the analysis model is the same as that of the conventional single-motion vibration absorber. Here, the damp pot damping coefficient is set as D _2eq, and the damping ratio of the dynamic vibration absorber 1 is defined by the following equation (55).

The optimum values of the natural frequency ratio f ₂ and the damping ratio γ _2eq between the main system 2 and the first dynamic vibration absorber (cantilever 3) are calculated using the following formulas (56), ( 57) (see Non-Patent Document 3).
Here, equations (56) and (57) are optimum values when the fixed point theory is applied with compliance as an evaluation index.

Using the equations (56) and (57), the optimum values of the spring constant K ₂ and the damping coefficient D _2eq are expressed as the following equations (58) and (59), respectively.

The analysis model of FIG. 5A in which the dashpot of FIG. 4 is returned to the original electric system and the mass point of the main system 2 is fixed in the space is equal to the passive vibration analysis model using the piezoelectric element 4 and the LR series circuit. In addition, the optimum value of the natural frequency ratio f ₃ between the first dynamic vibration absorber (cantilever 3) and the double dynamic vibration absorber (electric system including the piezoelectric element 4 and the LR series circuit) is The optimum value of the resistance ratio ζ of the heavy dynamic vibration absorber (electric system composed of the piezoelectric element 4 and the LR series circuit) is given by the following equations (60) and (61) (see Non-Patent Document 1).
Here, equations (60) and (61) are optimum values when the fixed point theory is applied to the compliance when an external force is applied to the mass point of mass M ₂ in FIG. 5A as an evaluation index.

From the condition that the maximum amplitude of compliance when using equations (60) and (61) and the maximum amplitude of compliance when the electric system is replaced with a dashpot, the equivalent damping ratio γ _2e of the electric system is defined, The following equation (62) is obtained.

Therefore, the equivalent attenuation coefficient D _2e of the electric system is the following equation (63).

The optimum equivalent stiffness ratio β _opt is approximately obtained as shown in the following equation (64) from the condition where the equivalent damping coefficient D _2e and the optimum damping coefficient D _2eqopt given by the equation (59) are equal.
Here, β _opt << 1.

From equation (64), it can be seen that the required equivalent stiffness ratio increases as the mass ratio of the dynamic vibration absorber 1 increases. Using the equation (64), the optimum value K _3opt of the spring constant K ₃ , the optimum value M _{3opt of} the mass M ₃ , and the optimum value D _{3opt of} the damping coefficient D ₃ are _expressed by the following equations (65) to (67), respectively. become.

Further, each of the optimum value L _opt of the inductance L the optimum value R _opt of the resistance value R of the following equation (68), obtained as (69).

<1.5 Numerical optimization of double dynamic vibration absorber>
The analytical optimal adjustment formula obtained in the previous section was derived using an approximation that replaces the electrical system with a dashpot, and is not completely optimal. In addition, the result of the previous section suggests that if the mass ratio μ is determined, the optimum value of the double dynamic vibration absorber (dynamic vibration absorber 1) is determined. Therefore, the optimum equivalent stiffness ratio β _opt , optimum natural frequency ratios f _2opt , f _3opt , and optimum resistance ratio ζ _opt obtained by numerical optimization under the condition of minimizing the maximum value of compliance are expressed by the following equation (70): -Shown in (73).

Here, a procedure for determining the specifications of the dynamic vibration absorber 1 from the optimum equivalent rigidity ratio β _opt , the optimum natural frequency ratios f _2opt and f _3opt , and the optimum resistance ratio ζ _opt will be described with reference to FIG. 3B. The mass M ₁ and the spring constant K ₁ of the main system 2 are determined in advance. First, M _{2 that} is substantially equal to the tip mass m of the dynamic vibration absorber 1 is determined based on the restriction due to the size of the dynamic vibration absorber 1. Therefore, when the optimum natural frequency ratio f _2opt is obtained, K ₂ is determined. Next, K ₃ is determined from the determined K ₂ and the optimum equivalent stiffness ratio β _opt . Further, M ₃ is determined from the determined M ₂ , K ₂ , K ₃ and the optimum natural frequency ratio f _3opt . Finally, D ₃ are determined from the optimum resistance ratio zeta _opt.

<1.6 Verification by optimal adjustment type simulation>
A simulation was conducted to verify the effectiveness of the analytical optimal adjustment formula obtained using fixed point theory in Section 1.4 and the numerical optimal adjustment formula obtained by numerical optimization in Section 1.5. FIG. 6A and FIG. 6B show the results of simulation of the dimensionless compliance magnitude when the mass ratio μ is 0.001, 0.01, and 0.1. FIG. 6A shows the result of using the analytical optimum adjustment formula, and FIG. 6B shows the result of using the numerical optimum adjustment formula. Tables 1A and 1B show the values of β _opt , f _2opt , f _3opt , and ζ _opt in each case. Although the difference between the value of the analytical optimal adjustment formula and the numerical optimal adjustment formula is small, the outline of the dimensionless compliance is different, and the results using the numerical optimal adjustment formula have higher damping performance. However, the difference in damping performance between the result of using the analytical optimal adjustment formula and the result of using the numerical optimal adjustment formula is about several dB, so when using the analytical optimal adjustment formula, the numerical optimal adjustment formula was used. Both cases may be considered optimal. When the analytical optimum adjustment formula is used, the vibration suppression performance is different between the low frequency side and the high frequency side. For example, it is suitable when it is necessary to suppress the vibration of the main system 2 where the vibration frequency distribution is not uniform. Yes. On the other hand, when the numerical optimum adjustment formula is used, the heights of the three peaks are almost the same, which is suitable when the vibration frequency distribution is uniform. The value may be adjusted from the optimum value (reference value) obtained from the numerical optimum adjustment formula according to the distribution characteristic of the vibration frequency. For example, when the vibration on the low frequency side is slightly large, the natural frequency ratio f ₂ may be adjusted to a value slightly smaller than the optimum value (reference value) obtained from the numerical optimum adjustment formula. Conversely, it may be adjusted to a slightly larger value than the optimum value determined natural frequency ratio f ₂ from the value optimum adjustment formula when the vibration of the high frequency side is slightly larger (reference value). Even when the analytical optimum adjustment formula is used, the value may be adjusted from a value (reference value) obtained from the analytical optimum adjustment formula in accordance with the distribution characteristic of the vibration frequency.

<< 2 Verification by experiment and simulation >>
FIG. 7 shows an outline of the experimental apparatus. In FIG. 7, the same parts as those in FIG. In the experimental apparatus shown in FIG. 7, a vise 9 is fixed on a vibration table 8 of the vibration apparatus 7, the base end of the main system 2 is fixed by the vice 9, and the base of the cantilever 3 is fixed to the front end of the main system 2. The ends are fixed. Further, the piezoelectric element 4 is attached to the cantilever 3, the acceleration sensor 10 is attached to the vise 9, and the acceleration sensor 11 is attached to the tip of the main system 2. Although the LR circuit connected to the piezoelectric element 4 is not shown in FIG. 7, an LR series circuit is actually connected between the electrodes of the piezoelectric element 4. The theoretical analysis in Chapter 1 considered the case of applying external force to main system 2 and applying vibration, but in the experiment, forced displacement was given. Displacement excitation at the lower end of the main system 2 can be considered equivalently by force excitation. Therefore, even in this experimental apparatus, compliance is measured by measuring the displacement transmission rate or acceleration transmission rate between the excitation table and the main system 2. Can be obtained (see Non-Patent Document 4). FIG. 8A and FIG. 8B show the magnitude of compliance obtained by simulation and experiment using the optimum value of the double dynamic vibration absorber (dynamic vibration absorber 1) obtained from the mass ratio μ of the experimental apparatus and the numerical optimum adjustment formula. Table 2 shows specifications of the double dynamic vibration absorber (dynamic vibration absorber 1) for simulation and experiment. Also in the experiment, the vibration can be greatly reduced by the double dynamic vibration absorber (dynamic vibration absorber 1) using the piezoelectric element 4, and the effectiveness of the dynamic vibration absorber 1 can be confirmed.

<< 3 Summary >>
In the present embodiment, a passive double dynamic vibration absorber (dynamic vibration absorber 1) using the piezoelectric element 4 has been proposed. It is theoretically shown that by attaching a piezoelectric element 4 to the leaf spring portion of the dynamic vibration absorber 1 and connecting an LR series circuit thereto, it can function equivalently as a double dynamic vibration absorber. Was used to derive the governing equation. Using an approximation that replaces the damping effect due to the electrical system with a linear dashpot, an analytical optimum adjustment formula for the double dynamic vibration absorber (dynamic vibration absorber 1) using the piezoelectric element 4 is obtained, and a numerical optimization method is used. Revised the optimal adjustment formula. The validity of the theoretical analysis was confirmed by simulation and experiment, and it was shown that the dynamic vibration absorber 1 is effective.

<< 4 Variation >>
In the above-described embodiment, the analytical optimum adjustment formula and the numerical optimum adjustment formula are obtained using compliance. However, the analytical optimum adjustment formula and the numeric value are obtained using other evaluation indexes (for example, acceleration, mobility, etc.). An optimal adjustment formula may be obtained.

  In the embodiment described above, the longitudinal direction of the dynamic vibration absorber 1 and the main system 2 are shown in FIG. 1A so that the movement of the dynamic vibration absorber 1 due to the movement of the main system 2 is the largest and the vibration absorption performance is the best. The longitudinal directions of these are matched. However, in practice, the longitudinal direction of the dynamic vibration absorber 1 and the longitudinal direction of the main system 2 cannot always coincide. However, even if the longitudinal direction of the dynamic vibration absorber 1 and the longitudinal direction of the main system 2 do not coincide with each other, the dynamic vibration absorber 1 can function as a bending vibration body type dynamic vibration absorber with damping. That is, for example, the longitudinal direction of the dynamic vibration absorber 1 may be inclined with respect to the longitudinal direction of the main system 2 in a plane perpendicular to the paper surface in FIG. 1A and including the dynamic vibration absorber 1. The longitudinal direction of the dynamic vibration absorber 1 may be inclined with respect to the longitudinal direction of the main system 2 in the plane, or both may be performed simultaneously.

In the above-described embodiment, some of the specifications (K ₂ , β, L, R) of the dynamic vibration absorber 1 are determined using the optimum value (reference value) obtained from the numerical optimum adjustment formula. A part of specifications (K ₂ , β, L, R) of the dynamic vibration absorber 1 may be determined from an optimum value (reference value) obtained from the analytical optimum adjustment formula. Further, the specifications of the dynamic vibration absorber 1 determined from the optimum value obtained by the numerical optimum adjustment formula or the analytical optimum adjustment formula are corrected in consideration of, for example, general-purpose product standards and costs of each component, and the corrected dynamic vibration absorber 1 These specifications may be the final specifications of the dynamic vibration absorber 1.

Further, by adopting the configuration of the dynamic vibration absorber 1 ′ shown in FIG. 9, it is possible to improve the attenuation effect by the electric system including the piezoelectric element 4 and the electric circuit. The dynamic vibration absorber 1 ′ shown in FIG. 9 has a configuration in which a vibration sensor 12 and an amplifier 13 are added to the dynamic vibration absorber 1 shown in FIG. 1A. The vibration sensor 12 is a sensor that detects the vibration of the cantilever beam 3 and outputs an electric signal corresponding to the displacement of the cantilever beam 3 or an electric signal corresponding to the acceleration of the cantilever beam 3 as a detection signal. The amplifier 13 amplifies the detection signal of the vibration sensor 12 and outputs it. Voltage v _a output from the amplifier 13 is applied to the LR series circuit. Thereby, compared with the dynamic vibration absorber 1, the voltage generated by the piezoelectric effect in the piezoelectric element 4 can be increased in a pseudo manner. As a result, the damping effect by the electric system composed of the piezoelectric element 4 and the electric circuit can be obtained. improves.

  Further, by adopting the configuration of the dynamic vibration absorber 1 "shown in FIG. 10, it is possible to reduce the damping effect by the electric system composed of the piezoelectric element 4 and the electric circuit. The dynamic vibration absorber 1" shown in FIG. In this configuration, a capacitor 14 is added to the dynamic vibration absorber 1 shown in 1A. In FIG. 10, the capacitor 14 is connected in parallel to the LR series circuit, but may be connected in series to the LR series circuit. By connecting the capacitor to the LR series circuit in this way, the voltage generated by the piezoelectric effect in the piezoelectric element 4 can be reduced in comparison with the dynamic vibration absorber 1, and as a result, the piezoelectric element 4 In addition, the attenuation effect due to the electric system including the electric circuit is reduced.

  In this way, the vibration damping performance can be easily adjusted by executing the deformation from the dynamic vibration absorber 1 shown in FIG. 1A to the dynamic vibration absorber 1 ′ shown in FIG. 9 or the dynamic vibration absorber 1 ″ shown in FIG. it can.

  In the above-described embodiment and its modification, the electric circuit connected between the electrodes of the piezoelectric element 4 includes the LR series circuit, but an LR parallel circuit may be used instead of the LR series circuit.

  The attachment position of the piezoelectric element 4 is preferably a position where the curvature of the fundamental vibration mode of the cantilever beam 3 is high, that is, a position close to the base end of the cantilever beam 3. It does not matter if it becomes a position.

  Further, in order to increase the strain energy stored in the piezoelectric element 4, for example, the piezoelectric element 4 is not directly attached to the cantilever beam 3, but via a spacer that supports both ends in the length direction of the piezoelectric element 4. A structure to be attached to the cantilever 3, a structure in which a notch is provided in the cantilever 3 and the piezoelectric element 4 is embedded in the notch, or a notch in the cantilever 3 is provided with the notch A configuration in which both ends in the length direction of the piezoelectric element 4 are attached to the cantilever 3 so as to be bridged may be employed.

  In the above-described embodiment, the single piezoelectric element 4 is attached to the cantilever beam 3. However, a plurality of piezoelectric elements can be attached and these can be electrically connected in parallel or in series. For example, when a plurality of piezoelectric elements are affixed to the cantilever 3, all the piezoelectric elements may be affixed to one side of the cantilever 3. Also good.

DESCRIPTION OF SYMBOLS 1 Dynamic vibration absorber 2 Main system 3 Cantilever 4 Piezoelectric element 5 Inductor 6 Resistor 7 Excitation apparatus 8 Excitation table 9 Vise 10, 11 Acceleration sensor 12 Vibration sensor 13 Amplifier 14 Capacitor

Claims (6)

A double dynamic vibration absorber attached to a vibration body to be controlled and suppressing vibration of the vibration body,
A first dynamic vibration absorber which is a bending vibration body having one end attached to the vibration body;
A double dynamic vibration absorber comprising a piezoelectric element and an electric circuit including an inductor and a resistor;
With
The piezoelectric element is attached to the bending vibrator and converts a load applied to the bending vibrator into a voltage.
The double dynamic vibration absorber is characterized in that the electric circuit is connected between electrodes of the piezoelectric element. It is a design method of the double dynamic vibration absorber according to claim 1,
Performing an approximation to replace a damping effect by an electric system composed of the piezoelectric element and the electric circuit with a linear dashpot;
Deriving a governing equation when the approximation of the double dynamic vibration absorber is performed using an equivalent mechanical model when the approximation of the double dynamic vibration absorber is performed;
A step of analytically obtaining an optimum adjustment formula of the double dynamic vibration absorber using an evaluation index obtained from a governing equation when the approximation is performed;
Determining a part of the specifications of the double dynamic vibration absorber using a reference value obtained from the optimum adjustment formula or an adjustment value obtained by adjusting a value from the reference value;
A method of designing a double dynamic vibration absorber, comprising: In the design method of the double dynamic vibration absorber according to claim 2,
The step of obtaining the optimum adjustment formula includes:
A sub-step for obtaining respective optimum values of the spring constant and the damping constant of the double dynamic vibration absorber when a fixed point theory is applied using an evaluation index obtained from a governing equation when the approximation is performed;
The inductor when the fixed point theory is applied to the analysis model in which the dashpot is returned to the original electrical system and the mass point of the vibrating body is fixed in space using the evaluation index obtained from the governing equation of the analysis model Substeps for obtaining respective optimum values of the inductance and the resistance value of the resistor;
A method of designing a double dynamic vibration absorber, comprising: In the design method of the double dynamic vibration absorber according to claim 2 or claim 3,
The method for designing a double dynamic vibration absorber, wherein the evaluation index is any one of compliance, mobility, and acceleration. It is a design method of the double dynamic vibration absorber according to claim 1,
Performing an approximation to replace a damping effect by an electric system composed of the piezoelectric element and the electric circuit with a linear dashpot;
Deriving a governing equation when the approximation of the double dynamic vibration absorber is performed using an equivalent mechanical model when the approximation of the double dynamic vibration absorber is performed;
A step of analytically obtaining an optimum adjustment formula of the double dynamic vibration absorber using an evaluation index obtained from a governing equation when the approximation is performed;
Correcting the optimum adjustment formula by numerical optimization with an evaluation index obtained from a governing equation when the approximation of the double dynamic vibration absorber is not performed as a predetermined condition;
Determining a part of the specifications of the double vibration absorber using a reference value obtained from the optimum adjustment formula corrected by the numerical optimization, or an adjustment value obtained by adjusting a value from the reference value;
A method of designing a double dynamic vibration absorber, comprising: In the design method of the double dynamic vibration absorber according to claim 5,
The evaluation index is one of compliance, mobility, and tolerance;
The method for designing a double dynamic vibration absorber, wherein the predetermined condition is a condition for minimizing a maximum value of the evaluation index.