## Abstract

Squeeze film damping in systems employing micro-plates parallel to a substrate and undergoing small normal vibrations is theoretically investigated. In high-density fluids, inertia forces may play a significant role affecting the dynamic response of such systems. Previous models of squeeze film damping taking inertia into account do not clearly isolate this effect from viscous damping. Therefore, currently, there is no simple way to distinguish between these two hydrodynamic effects. This paper presents a simple solution for the hydrodynamic force acting on a plate vibrating in an incompressible fluid, with distinctive terms describing inertia and viscous damping. Similar to the damping constant describing viscous losses, an inertia constant, given by *ρL*^{3}*W*/*h* (where *ρ* is fluid density, *L* and *W* are plate length and width, respectively, and *h* is separation distance), may be used to accurately calculate fluid inertia for small oscillation Reynolds numbers. In contrast with viscous forces that suppress the amplitude of the oscillation, it is found that fluid inertia acts as an added mass, shifting the natural frequency of the system to a lower range while having little effect on the amplitude. Dimensionless parameters describing the relative importance of viscous and inertia effects also emerge from the analysis.

## 1. Introduction

There are many applications that involve the normal vibration of beams or plates in the proximity of a surface. For example, in atomic force microscopy, micro-scale resonant cantilevers are employed to sense the contact or van der Waals force to map the topography of the substrate (Chen *et al.* 1995; Butt *et al.* 2005). The accelerometer, one of the most widely used micro-electromechanical systems (MEMSs), employs multiple pairs of parallel plates as electrodes of a variable capacitor (Samuels 1996). While these are systems that work in air, liquid-immersed MEMSs are emerging for a wide range of applications. For instance, micro-cantilever probes have been used as density and viscosity sensors for liquid media (Patois *et al.* 1999; Bergaud & Nicu 2000; Shih *et al.* 2001) and more efficient power-harvesting MEMSs employing variable capacitors with liquid dielectric have been recently proposed (Kempitiya *et al.* 2009). All these devices employ micro-plates or micro-beams vibrating in the vicinity of a fixed wall (substrate) and are subjected to damping from a film of incompressible liquid located in the gap between the plate and the wall.

When a beam or plate vibrates in a direction normal to a nearby substrate, the fluid between the plate and the substrate is squeezed, giving rise to resistance to the motion of the plate in the form of hydrodynamic lubrication-like forces. This phenomenon is known as squeeze film damping and the device is a squeeze film damper (SFD). A solution for the squeezing of a fluid between two plates by Stefan (1874) was one of the first applications of the Navier–Stokes equations, even pre-dating the Reynolds lubrication theory itself, which appeared in 1887 (Szeri 1998). The SFD is generally thought of, in the traditional tribology world, as a macro-device used to suppress unwanted vibrations in high-speed rotordynamic machinery such as aircraft engines and was a topic of considerable research in the 1980s (e.g. Pan 1980; Rogers 1984; Ku & Tichy 1987; Tichy 1987).

With the advent of micro-systems, the evaluation of fluid damping has become once again of high importance. In MEMSs, the hydrodynamic forces often constitute the largest source of parasitic losses. Hence, the SFD phenomenon greatly impacts on the resonance frequency, Q-factor, velocity and amplitude of the system and an accurate evaluation of fluid forces is critical in the design and optimization process of the MEMS devices.

Previous studies of SFDs in micro-systems were mostly focused on devices working in a compressible fluid, such as air (Starr 1990; Hosaka *et al.* 1995; Pan *et al.* 1998; Veijola 2004, 2009; Homentcovschi & Miles 2008; Mohite *et al.* 2008; Pandey & Pratap 2008). Since the density of air is small, most of these models do not take into account inertia effects that are negligible. However, the inertia effect becomes significantly important to liquid-immersed MEMSs, where it may greatly affect the dynamic response (Naik *et al.* 2003; Harrison *et al.* 2007).

Fluid inertia is neglected in classical lubrication theory. Existing modified squeeze damper models considering inertia provide complex expressions for the hydrodynamic force without clearly isolating inertia from viscous effects (Veijola 2004, 2009; Homentcovschi & Miles 2008; Mohite *et al.* 2008). For instance, in Veijola’s (2004) work, starting from an integrated form of the Navier–Stokes equations, a compact model considering both the inertia and viscous effect of compressible fluid under rarefied gas conditions was derived. However, the two effects of inertia and viscosity were not discussed separately at great length. Therefore, at present there is no simple way to evaluate and distinguish between these two hydrodynamic phenomena. This makes it difficult to assess their relative importance and the impact different device parameters have on each of these effects.

The objective of this study is to develop a simple expression for fluid inertia force and to investigate its effect (along with viscous damping) on the dynamic response of a micro-plate undergoing small oscillations. The plate under consideration is parallel to a substrate, elastically supported by a pair of thin cantilevers and subjected to hydrodynamic forces from a thin film of confined fluid. Fluid dynamic analysis is performed to solve for the hydrodynamic force under the assumption of small amplitude, normal vibrations. The hydrodynamic force is first determined using an exact similarity solution obtained for thin film flow at an arbitrary oscillation Reynolds number. This is found to agree well with a compact-form perturbation solution determined for small Reynolds numbers, which is ultimately used in the analysis in order to separately express viscous and inertia terms. The governing equation describing the motion of the plate is then solved and the solution is presented using non-dimensional parameters (*N*_{i} and *N*_{v}) associated with the two effects. The frequency response (magnification factor) is derived as a function of non-dimensional numbers to illustrate the response of the system for different regimes. At the end, a case study is presented to show how inertia could dominate over viscous damping when the fluid is changed from air to liquid (water).

## 2. Analysis

### (a) Governing equations

A schematic of the system under consideration is shown in figure 1. The system consists of a rectangular, rigid plate of length *L* in the *x*-direction along the film, and width *W* in the *z*-direction (into the plane of the paper). A thin film of fluid is confined between the plate and an infinite substrate. The flow is considered to be two dimensional (no variation in the *z*-direction), which is strictly true for *W* ≫ *L* but is an accepted approximation for *W* ∼ *L* (Szeri 1998). The plate has a single degree of freedom in the *y*-direction of squeezing. It has an effective mass *m* and is connected to the substrate by an elastic structure with effective spring constant *k*. In a laboratory reference system, the positions of the plate and the substrate are *y*_{p} and *y*_{s}, respectively.

Assuming that under an external excitation this system undergoes small vertical (normal) vibrations, the position of the substrate *y*_{s} can be expressed as
2.1
where *Y*_{s0} is the initial position of the substrate, *δ* is a small dimensionless parameter representing the oscillation amplitude (typically ≤0.1), *H*_{n} is the nominal gap (the gap in the absence of both the driving force and the plate effective mass) and *ω* is the excitation frequency. Following the conventional behaviour of linear, single degree-of-freedom systems, the displacement of the plate *y*_{p} is expected to have in-phase and out-of-phase components and is given by
2.2
where *Y*_{p0} is the static plate position and *C*_{s} and *C*_{c} are the currently unknown coefficients for the in-phase (sine) and out-of-phase (cosine) components of the plate motion.

From Newton’s Second Law, the equation describing the motion of the plate is
2.3
where *F*_{el} is the elastic force, *F*_{fl} is the force acted upon the plate by the fluid and *g* is the gravitational constant.

In order to solve equation (2.3), the hydrodynamic and the elastic force must be determined. The dynamic elastic force *F*_{el} is straightforward,
2.4
The hydrodynamic fluid force *F*_{fl} is the sum of viscous force *F*_{visc} and inertia force *F*_{iner}. Determination of these quantities follows the methodology of Tichy & Modest (1978) and Tichy & Winer (1977) and is explained in the next section.

### (b) Derivation of the hydrodynamic force

The derivation of the hydrodynamic force is carried out employing the complex variable formulation, which greatly simplifies the mathematical analysis. Here, the real part represents the physical quantity.

In the case of two-dimensional, incompressible squeezing of a Newtonian fluid between parallel plates, the separation gap between the plates (or film thickness), *h*, is given by
2.5a
where , and is a complex constant corresponding to the *C*_{s} and *C*_{c} of equation (2.2) (*C*_{r} = *C*_{c} and *C*_{i} = −*C*_{s}).

The associated fluid velocity of the confined thin-film liquid along the *y*-direction can then be expressed as
2.5b
The dynamic of the incompressible thin-film fluid can be described starting from the two-dimensional continuity and momentum equations,
2.6
2.7
2.8
where *v*_{x} and *v*_{y} are the fluid velocity components in the *x*- and *y*-directions (along and across the film, respectively), *p* is the fluid pressure, *η* is the viscosity and *ρ* is the density of fluid. For a thin film (*H*_{n} ≪ *L*), the following orders of magnitude for the velocities apply:
2.9
The unsteady term ∂*v*_{x}/∂*x* is of order *δLω*^{2}, while convective inertia terms such as *v*_{x}(∂*v*_{x}/∂*x*) are much smaller, of order *δ*^{2}*Lω*^{2}. Likewise, the *y*-momentum velocity terms of equation (2.8) are much smaller by order *H*_{n}/*L* than the corresponding terms of the *x*-momentum in equation (2.7). Under these conditions, the Navier–Stokes equations (2.7) and (2.8) reduce to
2.10
This governing equation is linear, the nonlinear convection terms being suppressed, and the pressure does not vary across the film as in boundary layer analysis.

As for boundary conditions, the no-slip conditions apply on the lower surface *y* = *y*_{s}, and the upper surface at *y* = *y*_{p} = *y*_{s} + *h*. On both surfaces *v*_{x} vanishes, but *v*_{y} is non-zero as fluid elements move in the normal direction attached to the vibrating plate. The fluid pressure is ambient (atmospheric) at the edges of the plate. Hence, the boundary conditions are
2.11
The variables *v*_{x}, *v*_{y} and *p* are complex, their real parts representing actual physical quantities. In the interests of simplifying the notation, the superscript tilde ( ∼ ) was omitted for the complex dependent variables.

In the manner of boundary theory, for equations (2.6) and (2.10) a similarity solution can be postulated (White 1974),
2.12
and
2.13
where *f* and *y** are the dimensionless similarity variables (dependent and independent, respectively). In a physical sense, one can show from taking the real and imaginary parts of equations (2.12) and (2.13) that the real parts *f*_{r} and *f*^{′}_{r} represent the velocity profiles (of *v*_{y} and *v*_{x}, respectively) at the instant of time when the surfaces are separated by the mean value *H*_{0} (the static gap between the plate and the substrate), and the squeeze velocity magnitude is *H*_{n}*δω*. The imaginary parts *f*_{i} and *f*^{′}_{i} represent the velocity profiles when the film thickness is the maximum or minimum, and the squeeze velocity is zero. It is relatively straightforward to show that equations (2.12) and (2.13) identically satisfy the continuity equation (2.6).

At this point, equation (2.12) is substituted into equation (2.10). Performing some minor manipulations, and arranging the terms in powers of *δ*, equation (2.10) can be re-written as
2.14
where *N*_{Re} is a time-dependent unsteady oscillation Reynolds number. Noting that the quantity in brackets is not a function of *x*, integrating and applying the pressure boundary condition of equation (2.11) the pressure is found as
2.15
while the hydrodynamic fluid force is then
2.16

Since pressure is not a function of *y*, equation (2.14) is differentiated with respect to *y* to obtain
2.17
with boundary conditions
2.18
Equations (2.17) and (2.18) can now be expanded in a series for *δ* ≪ 1. Noting that *δ* is embedded in *h* and *y**, and rearranging equation (2.17) in terms of an unsteady (oscillation) Reynolds number parameter, the above expressions become
2.19
and
2.20
Likewise, the fluid force becomes
2.21

An exact closed-form solution to equations (2.17) and (2.18) for arbitrary *N*_{Re0} can be obtained by standard methods of ordinary differential equations with constant coefficients (Boyce & DiPrima 2004). The solution methodology is straightforward, although the resulting expressions are complicated:
2.22
The force coefficient of equation (2.21) becomes
2.23

The behaviour of the similarity function *f* and its derivative *f*′ is shown in figures 2 and 3. Referring to equations (2.12) and (2.13), *f* and *f*′ are proportional to the velocities, *v*_{y} and *v*_{x}, respectively. By taking the real part of equations (2.12) or (2.13), one can find that the real component of *f* (figures 2*a* and 3*a*) represents instants of time *t* = 0,*π*/*ω*,2*π*/*ω*,… when the film thickness is the mean value *H*_{0} and the squeeze velocity is the maximum *H*_{n}*δω*. Similarly, the imaginary part of *f* (figures 2*b* and 3*b*) represents instants of time *t* = *π*/(2*ω*),3*π*/(2*ω*),5*π*/(2*ω*),… when the film thickness is the maximum/minimum value *H*_{0}±*δH*_{n}, and the squeeze velocity is zero. By an alternative interpretation, the real part represents the viscous solution in the absence of inertia, and the imaginary part represents the inertia solution in the absence of viscous effects. The effect of the unsteady Reynolds number *N*_{Re0} is also shown in figures 2 and 3. At *N*_{Re0} = 10, the profile of the viscous lubrication solution (*N*_{Re0} = 0) in figure 3*a* is virtually unchanged, but begins to flatten at *N*_{Re0} = 50. In figure 3*b*, the lubrication solution has zero velocity, just as the squeezing velocity is zero at this instant. As the Reynolds number increases, a complex structure of fluid flow reversals owing to inertia is exhibited, although the surfaces are stationary at this instant.

To avoid the complexity of this exact solution, a regular perturbation approach can be employed to derive simpler expressions for the hydrodynamic forces. Thus, the dependent variable *f* = *f*_{r} + i*f*_{i} has been expanded in power series of the oscillation Reynolds number *N*_{Re0}, forming a regular perturbation
2.24
Equation (2.24) is substituted back in equations (2.19) and (2.20). The resulting expressions are expanded in series, and the real and imaginary parts are taken. The governing equations become
2.25
with boundary conditions
2.26

In equation (2.25), each of the terms multiplying the powers of *N*_{Re0} must separately equal zero. By conventional methods for solving linear ordinary differential equations (Boyce & DiPrima 2004), *f*_{0} and *f*_{1} for equations (2.25) and (2.26) are determined as
2.27

The behaviour of the dimensionless force function *F*^{+} is investigated next, similar to equations (2.21)–(2.23). The perturbation solution for *F*^{+} turns out to be very simple,
2.28
Figure 4 compares the exact and perturbation solutions with the unsteady Reynolds number *N*_{Re0}. As above, the left side is the real component, which represents the instant when the film thickness is the mean value and the squeeze velocity is maximum, and the right side is the imaginary component, which represents the instant when the film thickness is maximum and the squeeze velocity is zero. The results are surprising. The perturbation solution is very accurate up to *N*_{Re0} ∼ 10, and is accurate to engineering approximation at *N*_{Re0} ∼ 50. The whole premise of perturbation solutions is that the perturbation parameter is much less than 1, so that each successive term of equation (2.24) is smaller than the preceding term. Therefore, while it is expected that the solution is accurate for *N*_{Re0} ≪ 1, fortuitously it is valid for *N*_{Re0} ∼ 10.

A comparison of the derivative of the similarity function *f*′ (proportional to tangential sliding velocities *v*_{x}) for the exact and perturbation solutions is shown in figure 5 at Reynolds number *N*_{Re0} = 50. The perturbation solution is just beginning to deteriorate, but still captures the basic physical features of the flow. At *N*_{Re0} = 10 (not shown), the exact and perturbation solutions are indistinguishable.

Substituting equations (2.5b) and (2.28) into equation (2.21), and performing some elementary manipulations, it is found that 2.29

Taking the real part of equation (2.29), and returning to the terminology of the MEMS system (i.e. substituting *y*_{p}−*y*_{s} for *h*), the result can be expressed as viscous and inertia contributions to the squeeze film force
2.30
and
2.31
The symbols *c*_{v} and *C*_{i} denote dimensionless viscous and inertia coefficients, which in the case of the parallel rectangular plates just derived are found to be *c*_{v} = 1 and *C*_{i} = 1/10. For other plate geometries, e.g. a circular disc, the coefficients *c*_{v} and *C*_{i} would have different values (Hamrock 1994).

### (c) Solution for the dynamic system

In order to solve equation (2.3), a series of algebraic manipulations is performed. Upon substituting equations (2.1), (2.2), (2.4) and (2.30) and (2.31) (with the appropriate sign) into equation (2.3), the following expression is obtained after cancelling the common factor *δH*_{n}:
2.32
For the above equality to hold, coefficients of the sine and cosine terms must be zero,
2.33
and
2.34
These equations can be simplified by introducing the following dimensionless parameters:
2.35
where *ω*_{0} is the vacuum resonance frequency of the plate-support system in radians per second. The symbols *N*_{i} and *N*_{v} separate the density and viscosity of the fluid and thus are associated with inertia and viscous effects, respectively. Substituting *N*_{i} and *N*_{v} into equations (2.33) and (2.34), the two equations are reduced to dimensionless forms as
2.36
and
2.37

The expressions for *C*_{s} and *C*_{c} are then found by solving the above algebraic equations,
2.38
and
2.39

## 3. Results and discussion

The resonance response of the system is determined from the magnification factor, (Thomson 1965), which is expressed here in terms of *N*_{v} and *N*_{Re},
3.1

*C*_{mag} is plotted in figure 6 as a function of *ω** for different values of *N*_{v} (gauging viscous effects) and *N*_{Re} (representing the relative importance of fluid inertia) to show how the response of the system changes in media with different viscosities as inertia forces become more important. As can be seen from figure 6, for weak viscous media (*N*_{v} = 0.1), the resonance peak shifts slightly (approx. 10%) towards lower frequencies with increasing *N*_{Re}; however, it does not depart far from the natural frequency. In comparison, for highly viscous media (*N*_{v} = 1), a similar relative increase in inertia produces a much more pronounced shift in resonant frequency (approx. 70%). Moreover, the peak of *C*_{mag} also drops more (20% compared with 10%) in highly viscous media with increasing inertia.

To better understand the effect of inertia, the response of a realistic micro-plate system oscillating in air (low viscosity and inertia) and water (highly viscous and significant inertia) is presented next. In this case study, the plate length and width are taken as 500 μm and the nominal gap between that plate and the substrate is considered to be 35 μm. The natural frequency of the system is chosen as 6.56 kHz (the same range as in Naik *et al.* (2003) and Harrison *et al.* (2007)); hence *ω*_{0} is 4.12×10^{4} rad s^{−1}. This resonant frequency may be achieved, for example, by taking the effective mass as 1.38 mg and the effective elastic constant as 2344 N m^{−1}.

The magnification factor of this system is plotted in figure 7 as a function of applied frequency *ω* for different media. The continuous curve in figure 7 shows the response in air (*ρ* = 1.2 kg m^{−3}, *η* = 1.9×10^{−5} Pa s). The resonant frequency is identical to the natural frequency of the system. The dashed-dotted line is plotted keeping the same density as that of air (equivalent to neglecting the effect of inertia), while changing the viscosity to that of water (hence *ρ* = 1.2 kg m^{−3}, *η* = 0.001 Pa s). In this case the resonant frequency of the system remains the same. However, the amplitude decreases significantly, consistent with classical results for single degree-of-freedom damped vibration systems (Thomson 1965). The dotted line shows the response in water when inertia effects are taken into account (hence *ρ* = 1000 kg m^{−3}, *η* = 0.001 Pa s). In this case, both the magnitude and the position of the magnification factor are distinct from that of air. Moreover, it is evident that at high density there is a shift (approx. 6%) in the resonant frequency towards lower frequencies, as observed experimentally (Naik *et al.* 2003; Harrison *et al.* 2007). However, fluid inertia has little effect on the magnitude of the magnification factor, which remains almost unchanged although the density increased by three orders of magnitude. Finally, to illustrate the relative importance of viscous damping and fluid inertia in different media, these forces are shown in figure 8*a*,*b*. As expected, viscous damping dominates over fluid inertia in air. On the other hand, the relative importance of the two changes in water at this *N*_{Re} (approx. 50), where inertia becomes dominant. All these are aspects that need to be taken into consideration in the development of MEMS devices working in high-density fluidic media.

## 4. Conclusion

A solution has been presented for the single degree-of-freedom dynamic system consisting of a plate supported elastically above a substrate, confining a viscous fluid, and undergoing small normal oscillations. In analysing the fluid dynamics problem, a compact perturbation solution for small Reynolds numbers was obtained and verified against an exact, closed-form similarity solution determined for any unsteady Reynolds number. The perturbation solution was found to accurately describe the hydrodynamic forces for unsteady Reynolds numbers of the order of 10. Thus, fluid inertia and viscous forces in such dynamic systems can be rigorously characterized in a simple and compact manner for the first time.

The hydrodynamic fluid forces determined via the perturbation method were included in the equation of motion for the rigid plate, along with the supporting elastic forces and gravity. The solution describing the dynamics of the system showed that the effect of fluid inertia was to decrease the resonant frequency of the system, consistent with recent experimental results. These findings are of importance to dynamic MEMSs subjected to small oscillations and significant fluid inertia in the *N*_{Re} < 50 regime.

## Acknowledgements

This work was supported by the US National Science Foundation under awards CMMI 0824788 and ECCS 0925733.

- Received April 22, 2010.
- Accepted August 31, 2010.

- This journal is © 2010 The Royal Society