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This paper starts with describing a method of deriving the resonance mode of a pendulum utilizing its equation of motion. Its resonance mode is that the particle accelerates toward the equilibrium position in the vertical plane including its fixed point. Therefore, a resonance mode can be derived by finding its equilibrium position and something to accelerate from its equation of motion. To derive the yaw resonance mode, the former paper transformed the equation of motion of vehicles into a structure that matches the equation of motion of the pendulum by assuming a special vehicle speed. This paper derives the yaw resonance mode at general velocities by modifying the variables that satisfy the configuration of the equation of motion derived in the former paper. As a result, this paper found that the yaw resonance equilibrium position is on the vehicle speed vector at the position where the side slip angle of the vehicle body is 0 in a steady state turning. This paper also found that something that accelerates towards equilibrium is the rear wheel. This acceleration of the rear wheels toward this extension is the restoration of yaw resonance.

In this paper, it is confirmed that natural frequency of isolated table is operated and that degeneration, which means that some natural frequencies have the same value, has a bad influence. Pneumatic anti vibration apparatuses (AVAs) support that loaded devices, i.e., XY stages, operate accurately. However, many strict design conditions are imposed on AVAs with six degrees-of-freedom (DOF) which are used on manufacturing floor. There is a risk that the natural frequency of each motion mode approaches, in other words, that isolated table falls into degeneration. This disturbs accurate operations of XY stages. In this paper, AVA with two DOF is used to simplify motion instead of six DOF. First, it is stated that mode control is used to pressure control. Next, position differential feedback (PFB) is proposed. PFB, which is used to the natural frequency control, is applied for each mode. This means that air spring stiffness is considered as each motion mode, and these are deleted or given. It is shown that the natural frequency of each mode can be controlled individually, by calculation, simulation, and experiment. Finally, the degeneration is reproduced to AVA with two DOF by PFB. At this moment, acceleration control which effects to nearby the natural frequency interferes with each other mode. From these results, conclusions are the following: degeneration should be avoided, and mode PFB is effective to control natural frequency of each motion mode. In addition, mode PFB accomplishes not only avoiding degeneration, but also lowering the natural frequency.

The roll motion of the vehicle has an effect on the yaw natural frequency. Because the yaw natural frequency formula considering this effect is the solution to a quartic equation, the expression of the formula is thought to be complex and incomprehensible. Accordingly, a seemingly comprehensible approximate formula was suggested in the former paper. In this approximation process, it was assumed that the cornering forces of the front and rear wheels are generated simultaneously. On the other hand, a later report have indicated that the yaw resonance when the vehicle drives at a certain speed has a cornering forces phase difference of 90 degrees between the front and rear wheels. Therefore this paper formulates the yaw natural frequency and yaw damping ratio assuming the phase difference to be 90 degrees. As results of that, qualitatively, the design variables that dominate the characteristic equation are appropriately included in the yaw natural frequency and yaw damping ratio formulas, and quantitatively, the approximation error is reduced. Consequently, these formulas are believed to be more appropriate than the previously proposed formulas. These formulas indicate that, when the load distribution ratio of the front wheels becomes larger, the yaw natural frequency decreases and the yaw damping ratio increases. In addition, this paper also indicates the scope of these approximate formulas quantitatively.

This paper deals with the impact of dropped hollow rubber spheres against a planar surface. Hollow silicon rubber spheres are used as the impact source of the measurement of the floor impact sound insulation of buildings, but the impact characteristics have not been investigated. In order to clear the impact characteristics and show a strategy of a development of another type of source, we have conducted some experiments, FEM simulation on LS-DYNA and a one-dimensional mass-and-spring model analysis. The experiment of the impact of 4 types of rubber spheres, two hollow spheres with an outer radius of 90 mm and two solid spheres with a radius of 60 mm and 40 mm, against a surface plate were conducted. As the FEM solutions corresponded qualitatively with the experimental results, the impact analysis was done based on the FEM results. The FEM solution shows the shock stress propagation in a hollow sphere has two steps; the ‘first period’ is the stress propagation in a solid thickness part similar in a solid sphere and the following period is that to the hollow part of sphere. The frequency response of the impact force shows ‘notch’ frequencies correspond to the natural frequencies of the hollow sphere of free-free boundary condition. This is the key point to analyze the impact characteristics of dropped hollow spheres. We investigated the relations between durations, natural frequencies of free-free boundary condition and notch frequencies of impact in relation to the thickness ratio of hollow sphere. We have presented a simple model to estimate the duration and the peak impact of dropped hollow sphere for an actual design.

In this paper, we present a shape optimization method for periodic microstructures to maximize a specified vibration eigenvalue of a porous macrostructure. The homogenized elastic moduli calculated by the homogenization method are applied to the macrostructure to connect the microstructures with the macro structure. The KS function is introduced to solve the repeated eigenvalue problem hidden in vibration eigenvalue optimization. The shape optimization problem subject to the volume constraint considering the microstructures is formulated as a distributed-parameter optimization problem, and the shape gradient function is derived by the Lagrange multiplier method and the adjoint variable method. The shape gradient function is applied as a distributed force to update the design boundaries of the unit cells of the microstructures by the H1 gradient method. The smooth boundary shapes obtained by the H1 gradient method are suitable for manufacturing with a 3D printer. In the numerical examples, the eigenvalues and the optimum shapes were compared changing the number of the domains of the microstructures in the macrostructure. As a result, the effectiveness of shape optimization method for microstructures aimed at maximizing the vibration eigenvalue of a macrostructure was confirmed.

In this paper, we present a non-parametric material orientation optimization method for tailoring the vibration eigenvalues of laminated shell structures composed of orthotropic carbon-fiber-reinforced plastic (CFRP) materials. Both of the design problems to maximize an arbitrary vibration eigenvalue and to maximize the gap between arbitrarily specified the vibration eigenvalues are formulated as distribution parameter optimization problems based on the variational method. We introduce KS function to solve the multiple roots problem latent in the vibration eigenvalue optimization problem. The sensitivity functions derived are applied to the H1 gradient method for material orientation to determine the optimal material orientation of each layer. The effectiveness of the proposed method are confirmed by the numerical results of a cantilever plate and a three-layer rectangular shell structure. The proposed method makes it possible to tailor the vibration eigenvalues of laminated shell structures without design parametrization while maintaining the smooth material orientation.

Tandem control is the control to drive one load with multiple motors. Its basic method is a common torque method. The common torque method is that the position signal and the rate signal from the motor 1 feedbacks to obtain the torque reference value of the motor 1, and the same torque reference value as the motor 1 is given to the motor 2.This method does not cause a motor current difference, but we found the vibration when a disturbance is applied is significant. In this paper, we propose a method that halves the vibration settling time when a disturbance is applied and does not cause steady-state current difference. The method is the average rate feedback method. The position feedback signal is the average position of each motor, and the rate feedback signal is the average angular velocity of each motor. The signal obtained by multiplying the rate difference between the motors by the rate difference control gain is passed through a high-pass filter, and the torque of each motor is corrected using this signal. As a result, in the average rate feedback method, the response of the rigid body mode can be tuned by the rate control gain, and the damping of natural vibration can be improved by the rate difference gain. The response and the damping of the natural vibration can be tuned separately. With the feedback of the rate difference through the high-pass filter, the steady-state current can be made equal between the motors. As another comparison, we also examined an individual rate feedback method that feedbacks angular velocity signals individually. In the individual rate feedback method, the response of the rigid body mode and the damping of natural vibration can be simultaneously improved by the rate control gain. However, current difference occurs between the motors. It is not good because one motor reaches the current limit first due to the current difference.

To develop turbomachinery blades with high reliability, it is indispensable to verify the natural frequency, the damping, the resonant stress, etc. under operating condition, and utilize these data in the mechanical design of a new blade. BTT (Blade Tip Timing) and the telemetry system have been widely used to measure the modal parameters of turbomachinery blades. In measurement of the modal parameters of blades in rotation, it is well known that the mistuning phenomena make the identification of modal parameters difficult. Namely, because, in a mistuned bladed disk, responses of many vibration modes are superimposed, it becomes difficult to correctly identify the natural frequency and the damping of each vibration mode. This is one of the critical issues to be overcome for developing blades with high reliability. Recently LSCF（Least-Squares Complex Frequency-Domain）method has been proposed as an effective method for identifying the modal parameters from complicated frequency responses. This method was also applied to derive the modal parameters of a bladed disk from the frequency responses expressed by travelling wave coordinates. In this paper, the validity of the LSCF method is verified by numerical simulations, where the modal parameters of mistuned bladed disks are identified from the frequency responses consisting of many vibration modes. In addition, the accuracy of the conventional half power method used in identifying the blade damping is researched.

Technology development to improve road induced noise is necessary with the market expansion of the electric motor drive vehicles, because road induced noise is noticeable under no engine noise condition. One of the effective ways to achieve good performance is to avoid the natural frequencies of vehicle bodies from dominant frequencies of the transmitted forces between a suspension and a vehicle body, which vary as the characteristics of road surfaces. For this background, this paper proposes a new method for shifting the natural frequency of coupled two vibration systems which are connected at multi-points by coupling springs assuming semi-active control. Firstly, based on the sub-structuring method, the relation between column vectors in the compliance matrix at connection points of two systems is analyzed to make clear the condition of resonance occurrence. This condition is equivalent that all column vectors of the compliance matrix at connection points exist in a hyperplane. Secondly, utilizing the independence of each coupling spring’s degree of freedom, simple equation to calculate a coupling stiffness is derived to satisfy the condition of resonance occurrence. Finally, in a feasibility study, the method was applied to two beams connected with three translational springs. It is showed that the proposed method can shift an initial natural frequency to target frequency with a simple calculation.

In the field of precision positioning such as semiconductor exposure apparatus, weak disturbance vibration from the floor influences processing accuracy. In recent years, as the precision of the device increases, the vibration allowance value becomes more severe, and performance improvement of the vibration isolator is required. In order to improve the performance, there are researchs reducing natural frequency by the frequency characteristics of the vibration transmissibility from the floor to the vibration isolator. Reducing natural frequency and decreasing stiffness are equivalent, and main target was stiffness generated by compressed air of air spring. In addition, rubber bellows of air spring is also targeted for further performance improvement. We already proposed a method combining the two stiffness separately decreasing method. However, decreasing stiffness simultaneously destabilized the device, then it could not be completely decreased. In this paper, as a new stabilization method, stiffness is completely decreased only in the operation frequency band. Since the low frequency has positive rigidity, the localization is maintained. We demonstrated the reducing natural frequency by the experimental result. When stiffness was completely decreased, the servo stiffness which had not been focused in the past became dominant in determining the natural frequency. Therefore, an appropriate design of the servo stiffness is required for further reducing natural frequency. To satisfy this requirement, we calculated the relationship between natural frequency and servo stiffness and gave an index for appropriate design.

This paper starts with describing a method of deriving the resonance mode of a pendulum utilizing its equation of motion. This resonance mode is that the equilibrium position of the pendulum locates the vertical plane including its fixed point and the mass accelerates in proportion to the distance from the equilibrium position to its mass. Further, the equation of motion of vehicles was converted to a form conforming to the equation of motion of the pendulum. As a result, it was found that the equilibrium position of yaw resonance is the extension line of the vehicle speed vector at the front wheel position. Moreover, it turned out that its rear wheel accelerates toward this extension line in proportion to the distance from this extension line to the rear wheel is the yaw resonance mode under a special condition. Finally, the step steering response was considered. At the moment of steering input, the mode of the yaw lead time constant appears, and then the yaw resonance mode becomes apparent. Hence, the yaw resonance is revealed in the latter half of the transient response. Therefore, it is considered that the yaw natural frequency is suitable as a metric of the latter half behavior of the transient response.

This paper investigates the free vibrations of double beams, consisting of upper and lower beams, which are discretely connected by N springs. In the theoretical analysis, the modal analysis approach is employed to determine the expressions for the natural frequencies and vibrational modes of the system. In the numerical calculation, two cases, Cases A and B, are examined. In Case A, the two beams are connected by a single spring, while in Case B they are connected by two springs. In Case A, when the two beams have identical materials and dimensions, as the spring constant K1 increases, the natural frequencies of the odd-order vibrational modes are constant because the two beams vibrate in phase in their vibrational modes and the connecting spring is not stretched. The natural frequencies of the even-order vibrational modes are increased with the increase of K1 because the two beams vibrate out of phase. When the spring is attached at the middle of the beams, the natural frequencies p4n-1 (n=1, 2, …) equal p4n. In addition, when K1 reaches the specific values K1,n in this case, a set of three natural frequencies satisfies p4n-2=p4n-1=p4n, and a magnitude relationship of the natural frequencies is switched when K1 crosses the value K1,n. In Case B, when two springs are attached in symmetry with respect to the midpoints of the beams, a set of two natural frequencies satisfies p2n=p2n+1 for the specific values of the spring constants even if the springs are not attached to the positions of the nodes of the independent beams. The validity of the theoretical analysis was confirmed by comparing the theoretical results with the results obtained by the FEM analysis for Case A.

Cymbals are percussion instruments that vibrate and radiate sounds when hit with a stick or when used in pairs. The sound radiated from a cymbal depends on its vibration characteristics. Cymbals are made through spin forming, hammering and lathing processes. The spin forming creates the domed shape of cymbals, determining the basic vibration characteristics. The hammering and lathing make specific sound quality adjustments by changing the vibration characteristics. In this paper, we focus on how the hammering affects the cymbal's vibration characteristics. The hammering produces many shallow dents over the cymbal's surface, generating residual stresses in it. These residual stresses change the vibration characteristics. We perform finite element analysis of the hammered cymbal to obtain its vibration characteristics. In the analysis, we use thermal stress analysis to reproduce the stress distribution and then with this stress distribution we perform vibration analysis. The results show that the effects of thermal load (i.e., hammering) vary depending on the mode: an increase or decrease in the natural frequency. As a result, the peak frequencies and their peak values in the frequency response function change.

Karman vortex shedding occurs when the gas passes through the duct with tube bank of the heat exchangers, such as gas heaters and boilers. Very high level sound in called “self-sustained tones” occurs due to the interference of the vortex and the sound field in the duct. In general, baffle plates are used to suppress the self-sustained tone. However, it is difficult to use them effectively, because insertion conditions have not been established. Then, perforated plates are installed in both sides of the duct to suppress the self-sustained tones. As a result, it was clarified that the perforated plate could suppress the self-sustained tone as expected. However this experiment was conducted using perforated plates with aperture ratio over 1% and the suppression effect appears in all of these aperture ratios. Wherefore the critical aperture ratio at which the suppression effect begin to appear could not be grasped in that experiment. This paper aims to clarify the critical aperture ratio and the effect of a cavity volume which is used with perforated plate on the SPL. Experiments were carried out to clarify these items. As a result, it could be obtained that the critical aperture ratio is about 0.25% and the influence of the cavity volume on SPL is a little. The optimum aperture ratio is concluded to be 4%.

A magnetic damper composed of a permanent magnet and a conducing plate has the advantage that the magnetic damper can generate a damping force with no physically contact. The conducting plate needs to be thickened in order to obtain a stronger magnetic damping force, since the magnitude of the magnetic flux produced by a permanent magnet is limited. However, it is known that the magnitude of the magnetic damping force is limited, even if the conducting plate of the magnetic damper has a large thickness. It has been considered that skin effect by eddy currents is the cause of this limited magnetic damping force. In addition, it is known that eddy currents have a very slight effect on the natural frequency of a structure with a magnetic damper. In this paper, the coil method considering inductances is proposed as one of the modeling methods for magnetic dampers consisting of a ring magnet and a conducting disk moving relatively in an axial direction. Applying this method, a modal analysis of the free and the forced vibration of a 1-DOF system with a magnetic damper are performed, and the magnetic damping and the magnetic stiffness considering the skin effect, are introduced. Furthermore, the experiments are performed in order to confirm the practicality of the method. The analytical results are found to be in agreement with the experimental results.

Dynamic model is necessary to predict the product performance in numerical simulation; strength of structure, noise and vibration. An experiment must be performed to identify the model parameter such as structural properties of installed or completed products because of preventing deterioration of prediction accuracy accompanying product dispersion and aging deterioration. Although several identification methods for elastic beams with frequency equation have been proposed in the previous studies. In these methods, the structural properties were derived from natural frequencies of the beam with/ without additional weights. The structural properties are not uniquely determined due to various factors such as boundary conditions and additional weights. Especially, no study deals with experimental conditions to enable the high prediction accuracy. This paper proposes novel methods to identify the structural properties of elastic beams of homogeneous, uniform cross-section in the lengthwise direction with free-free condition. In addition, the effects of controllable experimental parameters on the accuracy are examined. Two identification methods of the structural properties are derived; one is based on the frequency equation of the beam and the other is mass response method. It is necessary to measure natural frequencies of the beam with/ without additional weights on the beam to use both methods. The former can treat the change of mode shapes due to the additional weights, whereas the latter assumes that the mode shapes are unchanged. Finite element models are developed for validation of both methods. Various experimental conditions are adopted and the accuracy of the identified parameters by the both methods is compared. The mode shapes of the free-free beam are changed with additional weights. Thus, the method using the frequency equation is more applicable to the beam with the additional weights. Finally, the method based on the frequency equation of the beam is validated through the experiments.

This paper proposes a novel seismometer-type absolute displacement sensor aimed at detecting earthquake waves with a large magnitude and long period. However, since the measuring range of the conventional displacement sensor is higher than the natural frequency of itself, it is difficult to detect a low frequency vibration below 1Hz using conventional seismic-type displacement sensors. In order to provide an absolute displacement detection which is capable of lowering the natural frequency and enlarging the detectable amplitude without causing a structural defect, the relative signals of displacement, velocity, and acceleration between a detected object and the auxiliary mass of the sensor are fed back into the sensor. In addition, a phase lag compensation is inserted to adjust phase angles, which are below a frequency lower than 1Hz. Theoretically, this sensor is able to detect a lower frequency range from 0.2Hz to 10Hz and large amplitude with 100 times to the movable range of the sensor. It has been demonstrated by experiments that the developed sensor with a small size and lightweight has a detecting range from 0.2 Hz to 10Hz for absolute displacement and velocity. As an additional advantage, the measurement displacement amplitude has been expanded to about 40dB (100 times) for detecting earthquake waves with large magnitude and long period. As the typical application example of the developed absolute displacement sensor, a seismic wave based on Hachinohe wave is used with well observation results.

The primary method of improving steering dynamic response performance is expected to involve position control. However the driver uses not only the position of the steering wheel but also torque for control. It has been pointed out that particularly in regions that are close to straight-line driving, torque is the primary means of steering control. Therefore in order to achieve better high-quality dynamic response, it will be necessary to set higher natural frequencies and damping ratios for force control. A universal method for achieving this can be achieved by examining the symbolic expressions for these factors. Force control natural frequencies and damping ratios have been formularized for stable vehicles at all driving speeds. However these formulas cannot be used for vehicles which have unstable regions, and in fact there are vehicles which have such unstable regions. This paper examines a method of setting higher natural frequencies and damping ratios in order to improve the quality of dynamic response characteristics for vehicles that have unstable regions. I first envision a vehicle with neutral steering and steering system damping of 0, and confirm that the characteristic formula is a fourth-order equation for the Laplacian operator s. Next I show that when s is converted to a certain variable, the characteristic formula is written as a biquadratic equation for that variable. By solving this biquadratic equation, the damped natural frequencies and damping ratios are formularized. By considering these formulas, I show that increasing the cornering coefficient is a method that can simultaneously increase the damped natural frequency and damping ratio. I also show that this method can be applied to under-steer vehicles and vehicles which have steering system damping, and finally demonstrate the utility of this method with a time history response in transitional steering.

For characterizing vehicle dynamic behavior, the natural frequency and the damping ratio under position control, that is the control using steering angle as driver's input, have been developed as representative indices. However, a driver believes that he/she is controlling the vehicle maneuvers not only by steering angular input but also by steering torque input, that is, force control. Thus, in order to improve vehicle dynamic behavior further, it would be essential to carry out an analysis of the effect of force control on the natural frequencies and the damping ratios. However unlike position control, those indices under force control have been formulized only for neutral steer vehicles with a specific condition. The reason for this is that the characteristic equation of the system under force control consists of a combination of the steering system and the vehicle system, which turns into a biquadratic equation about Laplace operator s. To solve this equation, this paper formulized those indices by an approximation using general vehicle specifications, with final confirmation that the approximation errors do not exceed 5%. Finally, an example of an improvement in the vehicle dynamic behavior was demonstrated by comparing the results of original settings and those of assumed settings. Based on this result, it is shown that the vehicle dynamic behavior could be identified by force control indices as well as position control indices.

This paper describes the investigations of the characteristic about the sound generation of a nose flute experimentally. The nose flute is attached to the upper part of a container. If air is passed to a nose flute, sound will occur. The volume of a container is changed and the generated sound is measured. The natural frequencies of an experimental device are calculated and we confirm that it is in agreement with frequency of sound generated in experiment. We show that nose flute is a unique musical instrument with the point that a nose flute has only an edge part and uses people's mouth for a resonance body part. The frequencies of resonance sound can be calculated from the capacity in a mouth, the thickness and the area of an opening of a nose flute. When people play a nose flute, it is thought that only the first mode of vibration is used.

In this paper, the phenomenon of vibration and noise in a small fan motor caused by electromagnetic forces were experimentally investigated. First, vibration characteristics of the fan motor in a non-operating state were determined by experimental modal analysis. Next, vibration and noise characteristics of the fan motor in an operating state were determined. The rotational speed was continuously decreased from 2200 r/min to 1200 r/min while the measurements were taken. A detailed frequency analysis was also performed to investigate the directional patterns of noise at the rotational speed which the significant resonance were confirmed. In addition, analysis on the vibration and noise of the fan motor after turning off its power were performed. As a result, it was found out that the significant vibration and noise were generated in the axial direction, and were resonance phenomena between harmonic frequency of electromagnetic force and natural frequency of the fan.

The flatness of the metal thin plate shape is required in the production lines. Though, the convenient method for measuring small buckles of the thin plate has not been established. The tension distribution corresponding to the strain distribution is occurred by adding the strip running tension to the buckled thin plate. We propose an identification method by using natural frequencies and vibration modes, which are related to tension distributions of thin plate. In this paper, development of experimental equipment and applying the proposed identification method to measured vibration characteristics of a thin aluminum plate is investigated. Then, it shows the results of comparison between the identified tension distribution and directly measured value by the strain gauges. As a result, it is shown that identified tension distributions well corresponded to the values measured by strain gauges.

Vortices occur when gas flow passes through a duct with a tube bank of the heat exchangers, such as a gas heater and a boiler. Very high level sound called a “self-sustained tone” occurs due to the interference of the vortices and the sound field in the duct. There are many investigations concerning the generation mechanism and countermeasures of the self-sustained tones. However there are few investigations focusing on the behaviors of the pressure and the velocity in occurrence of the self-sustained tones. In the past, we experienced the failure of the fan blade used in the experiment. In this paper, the experiment was carried out to examine the behaviors of the pressure and the velocity in the duct with tube bank. As a result, the static pressure changes along the squared curve when the self-sustained tone does not occur. But it suddenly rises up when the self-sustained tone occurs and it is possible for this pressure rise to cause the blade failure due to the generation of the rotating stall. In addition, it was clarified that the flow velocity does not increase in proportion to the inverter frequency and its gradient decreases when the self-sustained tone occurs.

Road noise is one of the main sound sources exciting passenger car interior noise. This noise is excited by road roughness. Vibration and noise characteristics in road noise have strong interaction with tire vibration characteristics. Car manufactures are keenly interested in studies on the prediction of NVH (Noise, Vibration and Harshness) performance, including viewing tire as substructure. Tire vibration has complex behavior due to tire contact with the road and rolling. In this paper, we focus on the effect of contact between tire and road as against tire vibration. First, experimental modal analysis is performed to grasp the tire vibration on the non-contact and contact condition. As a result, it was clarified that natural frequency and mode shapes change on the contact condition. Second, we studied the factor of changing tire vibration characteristics between contact and non-contact using a tire model. A tire on the non-contact condition was expressed as a shell model. Based on the tire model, we performed vibration analysis using the reseptance method and definition of eigen function. From this study, it was found that prediction accuracy by receptance method depends on the number of approximate modes. The best method for determing the number of approximate modes is based on the experiment. Additionally, analysis method by definition of eigen function also is valid sufficiently.

This paper presents a non-parametric, or a node-based, shape optimization method for designing the optimal geometry of a space frame structure for a natural frequency problem. For a frame structure composed of arbitrarily curved linear elastic members, a natural frequency maximization problem for a specified mode subjected to a volume constraint is formulated as a distributed-parameter shape optimization problem. With the eigenvalue of a specified mode as the objective functional and the assumption that each frame member varies in the out-of-plane direction to the centoroidal axis, the shape gradient function and the optimality conditions are theoretically derived by the Lagrange multiplier method and the material derivative method. An optimal geometry is determined by applying the negative shape gradient function as fictitious external forces to the frame members and analyzing an optimal variation that minimizes the objective functional. This methodology was developed by the authors as the free-form optimization method for frame structures, which is a gradient method in a Hilbert space. For achieving the eigenvalue of a specified mode to be maximized, multiple root problem is avoided by tracking the specified mode through the optimization process using Modal Assurance Criterion (MAC). The effectiveness of the proposed method were verified through several design problems.