^{*}

A novel vibration isolator is constructed by connecting a mechanical spring in parallel with a magnetic spring in order to achieve the property of high-static-low-dynamic stiffness (HSLDS). The HSLDS property of the isolator can be tuned off-line or on-line. This study focuses on the characterization of the isolator using a finite element based package. Firstly using the single physics solver, the stiffness behaviours of the mechanical and magnetic springs are determined, respectively. Then using the weakly coupled multi-physics method, the stiffness behaviours of the passive isolator and the semi-active isolator are investigated, respectively. With the found stiffness models, a nonlinear differential equation governing the dynamics of the isolator is solved using the time-dependent solver. The displacement transmissibility ratios of the isolator are obtained. The study confirms that the isolation region of the isolator can be widened through off-line or on-line tuning.

Vibration isolation is one of the vibration suppression methods [1-5]. There are two kinds of vibration isolation: to isolate a vibration source from its support and to isolate a base excitation from a device. This study focuses on the second type. A simplest vibration isolator consists of a spring and a damper arranged in parallel. The spring plays a dual role: to support the weight of the device and to isolate the base motion. For a linear spring isolator, vibration isolation occurs when the natural frequency ω_{n} of the isolator system is smaller than where ω_{b} is the frequency of the base motion or exciting frequency. To increase the isolation region, the stiffness of the isolator spring should be made as low as possible. However, lowering the isolator’s stiffness results in a large static deflection that is undesirable. To overcome this problem, a high-static-low-dynamic stiffness (HSLDS) spring can be used. The HSLDS isolator is capable of supporting a large static load while possessing a low natural frequency [6,7].

A passive isolator is effective if operating conditions remain unchanged. However, if operating conditions vary, the isolator’s performance may deteriorate. A semiactive or tunable vibration isolator is capable of altering its stiffness or damping level in real time based on the response. Semi-active vibration isolators have been receiving increasing interest because they combine the advantages of both the passive and active devices [8-10]. On one hand, they preserve the reliability of the passive devices even in the event of power loss; on the other hand, they possess the adaptability of the active devices without a great amount of power consumption.

The tunable HSLDS isolator proposed in [11,12] belongs to the family of variable stiffness devices. The notable feature of the device is that its stiffness can be instantly decreased or increased without any moving parts. The ability of on-line tuning makes the isolator quickly adapt to the operating conditions so that the optimal performance is maintained. The isolator spring consists of a mechanical spring and an electromagnetic spring. As both the springs are nonlinear, determination of their stiffness models presents a challenge. In [11,12] an experimental approach was taken for this purpose. Another effective method to deal with such a system that involves two physics is to use a finite element (FE) based multiphysics software. This paper reports a characterization study for the isolator using Comsol Multiphysics, a commercial FE package. The aim of the study is twofold: to determine the stiffness behaviors of the isolator springs, and to evaluate the performance of the isolator in a passive mode or semi-active mode. The rest of the paper is organized as follows. In Section 2, the proposed HSLDS isolator is introduced. In Section 3, the stiffness characterization is addressed. In Section 4 the displacement transmissibility of the HSLDS isolator is evaluated. In Section 5, the study conclusions are presented.

As shown in _{b}. The interaction between the PM and the EMs results in an electromagnetic spring denoted as k_{pe}. The electromagnetic spring k_{pe} consists of two parts: k_{pc} and k_{pf} where k_{pc} is due to the interaction between the PM and the EM cores and k_{pf} is due to the interaction between the PM and the flux generated by the EMs. Note that the net magnetic force acting on the mass or PM behaves in the following way. When the mass is located exactly in the middle of the gap of the EMs, the net magnetic force is zero. With the EM polarities shown in

the mass moves to the left, the net magnetic force acts leftward, which corresponds to a negative magnetic stiffness. With the reversed EM polarities, when the mass moves to the left, the net magnetic force acts rightward, which corresponds to a positive magnetic stiffness. It should also be noted that the static stiffness depends only on the mechanical spring, while the dynamic stiffness comes from the combined effect of the mechanical spring and the electromagnetic spring.

To find the stiffness of the mechanical spring, a vertical force is applied to the center of the mass. Using the parametric solver, the displacements of the mass corresponding to a given range of the vertical forces are obtained. The solid line in

values computed from the force-displacement relationships. It is noted that the stiffness increases with an increase of the mass displacement. This indicates that the mechanical spring exhibits the characteristics of a hardening spring.

^{−1}. Each of the EMs is modeled as an assembly of a solid cylinder representing the steel core and a hollow cylinder representing the coil. The diameter and length of the solid cylinder are 13.0 mm and 150.0 mm, respectively. Its relative permeability is 152.7 when the EM current is zero. The inner and outer diameters of the hollow cylinder are 13.0 mm and 46.6 mm, respectively. The length of the hollow cylinder is 88.0 mm. The EM-PM-EM assembly is surrounded by two cylindrical free spaces. The dimension of the inner free space is 660.0 × 160.0 mm (length × diameter) and the dimension of the outer free space is 700.0 × 200.0 mm (length × diameter). The outer free space consists of infinite elements [

Magnetic force simulations require a fine mesh in order to obtain a reasonable accuracy. In particular, the mesh on the exterior boundaries of the object where the force is evaluated needs to be extra fine. A finer mesh reduces the computation error but consumes more computer resources. The “Normal” mesh was used in the model. In order to ensure an accurate evaluation of the force, the maximum element size is specified for the relevant members. In particular, the maximum element size of the two hollow cylinders is 4.0 mm. The maximum element size of the two pole faces of the PM and the adjacent end faces of the steel cores is 1.0 mm.

The magnitude of the external current density that flows in the cross section of the hollow cylinder is approximated to be the EM current density in each turn.

where J_{0} is the magnitude of the external current density, I is the current in the EM and r is the radius of the coil wire. The x component of the external current density is (the coordinate system is defined in

and the z component of the external current density is

When the PM is placed between the EMs that are not energized, the interaction between the PM and the steel cores of the EMs constitutes a permanent magnet spring. When the PM is in the middle of the gap of the EMs, the net force on the PM is zero. When the PM is displaced along the y axis, the net force tends to move the PM away from its equilibrium position. Such a force is considered to be a negative restoring force. By placing the PM at different locations along the y axis, the relation

ship between the net force and the PM displacement is obtained.

When the EMs are energized, the force acting on the PM depends on several factors such as the flux density produced by the EMs, the magnetization and position of the PM, and the relative permeability and size of the steel cores. Many studies have shown that the relative permeability of ferromagnetic materials is sensitive to several factors such as temperature and applied magnetic field intensity [14-16]. For example, when an electromagnet is energized, the temperature of its core will rise, which inevitably alters the relative permeability of the core. And when a PM is near a ferromagnetic material, the strong magnetic field generated by the PM will lower the relative permeability of the material. Therefore, selecting a proper relative permeability for the steel core plays a critical role in simulation.

In this study, to simplify the simulation, it is assumed that the relative permeability of the steel core depends only on the amplitude of the EM current. To obtain proper relative permeability values, the following approach was taken. First an experiment was conducted to measure the interacting forces between the PM and one EM. After a DC current was applied to the EM, the forces acting on the PM were measured for different gap distances between the PM and the EM. The circles in

rent, the relationship between the computed force and the gap distance was obtained. The root-means-squared (RMS) error between the computed force values and the measured ones for a given current was found. Varying the relative permeability value changed this RMS error. The simulation was repeated by varying the relative permeability values continuously until a minimum RMS error was obtained. The relative permeability value that resulted in a minimum RMS error was taken as the estimated value. The solid lines in

As pointed out previously, one of the main factors affecting the relative permeability is the temperature change in the core. An experiment was conducted to measure the temperatures of the core while the EM current varied. In the experiment, after a given current was applied to the EM for 30 minutes, the surface temperature of the core end was measured using a digital temperature meter. After that, the EM was let to cool down

for 30 minutes and the experiment was repeated for a new current.

With the estimated relative permeability values, the model of

When the mass-beam assembly is placed between the electromagnets that are not energized, a passive HSLDS isolator is obtained. To simulate the passive HSLDS isolator, a multiphysics functionality is required as it involves two physics, namely, structural mechanics and electromagnetism. Comsol software provides two ways to tackle a multiphysics problem: fully coupled and weakly coupled. In order to simplify the simulation, a “weakly coupled” way was employed in this study. This method solves for one type of physics at a time and then uses that solution as the initial value to solve for the other type of physics [

As shown in

where F_{b} is the restoring force of the beam and F_{m} is the net magnetic force. Both F_{b} and F_{m} depend on the location of the mass. When the model shown in _{b}. Therefore, by applying a force to the mass, the displacement of the mass can be found. Then the mass position in the model of _{m}. With F_{b}_{ }and F_{m}, the total restoring force F is obtained using Equation (4). This process was automated through a Comsol script.

sponding stiffness curves. It can be observed that reducing the gap distance will lower the isolator stiffness. When the gap distance is too small, for example D = 60.0 mm, the isolator stiffness in the neighbourhood of the equilibrium position becomes negative, resulting in an unstable system.

When the electromagnets are energized, the isolator’s stiffness becomes on-line tunable. The weakly coupled approach was again used to determine the total restoring forces of the isolator spring.

The previous section has shown that the dynamic stiffness of the isolator can be reduced by either reducing the

gap distance or by increasing the EM current. In this Section, the performance of the isolator is investigated by evaluating the displacement transmissibility ratio (T.R.). First the T.R. is defined. When a vibration isolator is subjected to a sinusoidal base excitation given by

where Y and ω_{b} are the amplitude and frequency of the base motion, respectively. The steady-state response of the isolated mass is given by

where X and θ are the amplitude and phase of the steadystate response of the isolated mass, respectively. The displacement transmissibility ratio (T.R.) is defined as

when T.R. < 1, the vibration isolation occurs.

Because the isolator stiffness is nonlinear, a closedform solution for its T.R. does not exist. In this study, a numerical solution was conducted using Comsol multiphysics package. Several attempts were made. One attempt was to use the weakly coupled method. It failed due to an enormous computational burden and convergence problem. In another attempt, the magnetic spring was treated as a lumped-parameter element whose restoring force equations were determined from the results presented in Section 3.2. This lumped-parameter spring was added to the model shown in

where m is the mass, c is the damping coefficient, z = x − y is the relative displacement of the mass, and f(z) is the restoring force of the isolator spring. For the passive isolator, the restoring force is obtained by adding the restoring force of the beam (solid line in _{b}t) as the base motion and using z = x − y, Equation (8) is rewritten as

The above equation was imported in the global equation dialog box in the Structural Mechanics module. The time-dependent solver was employed to solve the equation.

The damping coefficient was estimated by the following procedure. First, a free response experiment was conducted on the passive isolator. In the experiment, the mass was displaced by 2.26 mm and released. The induced free response was measured. Using the measured free response, the damping coefficient was estimated to be c = 0.13 Ns/m. Then a Comsol simulation was conducted with c = 0.13 Ns/m as a trial damping coefficient. The simulated response was compared with the measured one. The simulation was repeated by varying the trial damping coefficient values. The damping coefficient that resulted in the best agreement between the simulated response and simulated one was chosen to be the value used in the following simulations.

The amplitude of the sinusoidal base motion was taken as 1.0 mm and the initial conditions were zero. It was observed that the response became almost steady state after 24.0 seconds. A Comsol script was used to compute the RMS value of the response in the duration of 24.0 second and 30.0 second. The transmissibility ratio was computed using the ratio of the RMS value of the mass displacement over that of the base motion. By varying the exciting frequency from 1.0 Hz to 25.0 Hz in a step size of 0.5 Hz, the relationship between the transmissibility ratio and the base exciting frequency was obtained.

with the mechanical spring only (System 1), D = 64.0 mm (System 2), D = 68.0 mm (System 3), and D = 76 mm (System 4). Several observations can be drawn. With the mechanical spring only or System 1, the transmissibility ratio increases with an increase of the exciting frequency up to about 18 Hz. After that, the transmissibility ratio drops drastically. A further increase of the exciting frequency makes the value of the transmissibility ratio less than one. A sudden jump in the amplitude of the steady-state response is a typical phenomenon in systems with a hardening or softening stiffness [7,17]. In this paper, the lowest exciting frequency that causes such a jump phenomenon is referred to as the jump frequency. With the influence of the PM spring (Systems 2, 3, 4), the pattern of the transmissibility ratio curve remains similar but the jump frequency shifts to the left when the gap distance decreases. This indicates that the introduction of the PM spring widens the vibration isolation region. It is also noted that the PM spring causes an increase of the transmissibility ratio in the low frequency region.

The same process was employed to compute the transmissibility ratios for the semi-active isolator with a gap distance of D = 76 mm.

The characterization study for a novel vibration isolator

has been conducted using Comsol Multiphysics package. The stiffness behaviors of both the mechanical spring and magnetic spring have been determined. The study has shown that the mechanical spring possesses a positive nonlinear stiffness while the magnetic spring possesses a negative nonlinear stiffness. The simulation has shown that the stiffness of the magnetic spring can be varied by either changing the gap distance or the current. Using the weakly-coupled multiphysics approach, the combined stiffness of the passive isolator has been investigated. It has been found that the HSLDS property of the passive isolator can be tuned by varying the gap distance of the EMs. Using the weakly-coupled multiphysics approach, the combined stiffness of the semi-active isolator has been investigated. It has been found that the HSLDS property of the semi-active isolator can be varied by varying the EM current. A lumped-parameter model has been used to simulate the dynamic responses of the passive isolator and semi-active isolator to a base excitation. The obtained relationships between the transmission ratio and the exciting frequency have verified that the isolation region of the isolator can be effectively widened by either reducing the gap distance or increasing the current of the electromagnets. This study has demonstrated that a multi-physics finite element package can be used to tackle a nonlinear system that involves two different physics. The study has also shown that an experimental calibration or validation is critical in order to obtain a reliable computer simulation using a finite element based package.