Measurement of added mass for an object oscillating in viscous fluids using nonlinear self-excited oscillations

Abstract

In this study, a method of measuring the added mass for an object oscillating in a viscous fluid using nonlinear self-excited oscillations was developed. The added mass produces an additional inertial effect on the vibrating object. In previous methods, the added mass is obtained experimentally from the response frequency and amplitude at the peak in the frequency response curve under a harmonically forced excitation. However, such methods cannot be utilized in highly viscous environments because the peak becomes ambiguous as a result of damping. Moreover, in very high-viscosity environments, the resonance peak ceases to exist in the frequency response curves. To solve this problem, self-excited oscillations were induced in the system using linear velocity feedback. Because linear velocity feedback can be used to eliminate the viscous damping effect when the linear feedback gain is set near the Hopf bifurcation point related to self-excited oscillations, the object becomes self-excited at a frequency equal to the natural frequency of the object oscillating in a vacuum, i.e., the undamped natural frequency. What distinguishes the proposed method from the abovementioned previous methods is that the value of the response is not needed; however, a reduction in the oscillation amplitude is required to obtain accurate measurement results. In the proposed method, to avoid an increase in the response amplitude under the applied linear velocity feedback, nonlinear feedback is also applied to produce a limit cycle similar to that produced in a van der Pol oscillator. A prototype of the measurement system was constructed based on the proposed method. A comparison of the experimentally and theoretically obtained added mass values confirmed the validity of the proposed method for added mass sensing.

Appendices

Appendix A

Fig. 6

Circular plate oscillating in a viscous fluid

In this appendix, the closed solution of the added mass for a circular plate oscillating in a viscous fluid is derived. Fig. 6 shows a schematic of the plate according to Fig. 1. The motion of the viscous fluid is shown in this figure. When the Reynolds number of the fluid is very small, the Navier–Stokes equation can be simplified as

$$\begin{aligned} \frac{\partial u(y,t)}{\partial t}=\frac{\mu }{\rho }\frac{\partial ^2 u(y,t)}{\partial y^2}. \end{aligned}$$

(34)

The velocity of the fluid along the y-axis is governed by Eq. (34), where the $\mu$ and $\rho$ are the viscosity and density of the fluid, respectively. The case of a sinusoidal displacement of the plate was considered here. The displacement in this case is given by

$$\begin{aligned} x=a\mathrm{e}^{i\omega t}, \end{aligned}$$

(35)

where i is the imaginary unit and a and $\omega$ are the complex amplitude and frequency, respectively. The velocity and acceleration of the plate are then, respectively, expressed as

$$\begin{aligned} \dot{x}= & {} ia\omega {e}^{i\omega t},\nonumber \\ \ddot{x}= & {} -a\omega ^2{e}^{i\omega t} \end{aligned}$$

(36)

It was assumed that interfaces between the fluid and the plate and wall are no-slip interfaces. Hence, the fluid velocity is zero at the wall and equal to the excitation velocity at the plate. Thus, the boundary conditions of the velocity of the fluid at $y=0$ and $y=h$ are expressed as

$$\begin{aligned} u(h,t)=0, u(0,t)=\dot{x} = ia\omega \exp (i\omega t), \end{aligned}$$

(37)

where h is the distance from the wall of the container to the plate. With the above boundary conditions, u(y, t) is expressed as

Fig. 7

Motion of plate: a Vibration as a rigid body in x-direction, b Elastic vibration in y-direction

Fig. 8

FFT analysis of the motion of plate: a Vibration as a rigid body in x-direction, b Elastic vibration in y-direction

$$\begin{aligned} u(y,t)=\frac{ia\omega ({\mathrm{e}^{\gamma [2h(1-i)-y(1+i)]}}-\mathrm{e}^{\gamma (y-2h)(1+i)})}{2(\mathrm{\cosh }2\gamma h-\mathrm{\cos }2\gamma h)}\mathrm{e}^{i\omega t},\nonumber \\ \end{aligned}$$

(38)

where $\gamma =\sqrt{\frac{\omega \rho }{2\mu }}$. From Newton’s law of viscous flow, the fluid force along the x-axis is

$$\begin{aligned} F_f =-\alpha (b) S\sqrt{\frac{2\mu \rho }{\omega }}\ddot{x}-\beta (b) S\sqrt{2\mu \rho \omega } \dot{x}, \end{aligned}$$

(39)

where $\alpha (b)$ and $\beta (b)$ are expressed as

$$\begin{aligned} \alpha (b)&=\frac{\sinh b-\sin b}{\cosh b-\cos b},\nonumber \\ \beta (b)&=\frac{\sinh b+\sin b}{\cosh b-\cos b}, b=2\gamma h \end{aligned}$$

(40)

The first term on the right-hand side of Eq. (39) expresses the inertial force, which is what causes the added mass phenomenon. Thus, the theoretical value of the added mass is derived as

$$\begin{aligned} \delta m=\alpha (b) S\sqrt{\frac{2\mu \rho }{\omega }}. \end{aligned}$$

(41)

Appendix B

We measured the vibration in the y-direction at a point near the center of the plate (diameter: 76 mm) which is self-excited at a frequency of 12.9 Hz in the x-direction (Fig. 4). The experimentally obtained time histories in the x- and y-directions are plotted in Fig. 7a and b, respectively. Figure 8a and b presents the associated FFT analysis. The main frequency component corresponds to the natural frequency of the system given by Eq. (4), i.e., a rigid plate subjected to forces from a spring and fluid, but not to the natural frequencies associated with the elastic deformation of the plate. The amplitude of the oscillation in the y-direction was much smaller than the diameter of the plate (76 mm) and the amplitude of the oscillation as a rigid body in the x-direction. This characteristic probably holds for the other plates (diameters: 80 mm and 100 mm) used in the experiments described in this paper. Therefore, because the plates can be regarded as rigid bodies in the x-direction without elastic deformation in y-direction, these plates are suitable to confirm experimentally the validity of the proposed method, which is to measure the added mass for rigid plates oscillating in viscous fluids.