Research on nonlinear inertance calculation method of ball-screw inerter

3.1. Establishment of coordinate system

In order to carry out a dynamic analysis on the ball screw assembly in inerter, a World Coordinate System ($O$-$XYZ$) is established as shown in Fig. 2, in which the $Z$ axis coincides with the central axis of the screw; Moreover, a Frenet frame is established based on the motion curve of the ball center to study the sliding phenomenon of the contact area between balls and raceway.

Fig. 2Establishment of coordinate system

Set the base vector of the World Coordinate System as:

And the base vector of the Frenet frame as:

where the basic vector of the Frenet frame can be expressed as [21]:

The curve equation of the ball center in the World Coordinate System can be expressed as:

where $r_o$ represents the nominal radius of ball screw assembly, $\theta$ represents ball center track which is relative to the World Coordinate System turning angle, $\alpha$ represents spiral angle of the ball screw assembly.

To figure out the ratio between World Coordinate System and Frenet frame, take the derivative of Eq. (6) to obtain:

Substitute Eqs. (6)-(8) into Eq. (3)-(5) and get:

where define:

According to Eq. (9) and (10), one can know that:

And then by performing the matrix inverse of Eq. (10), one can obtain:

3.2. Contact point movement on ball

As shown in Fig. 3, in the Frenet frame, $A$ can be defined as the contact point between the ball and the nut, while ${\beta }_A$ as the contact angle, and $B$ can be defined as the contact point between the ball and screw, while ${\beta }_B$ as the contact angle.

Fig. 3Schematic of contact point between ball and raceway

Transforming Eq. (7) into the Frenet frame:

Assume that the instantaneous rotation angular velocity of the ball in the Frenet frame is as: ${\omega }_{ball}=\{{\omega }_t,{\omega }_n,{\omega }_b\}F\text{,}$ then the instantaneous velocity of contact point $A$ and $B$ can be denoted as:

In the two equations above, $r_b$ represents the ball radius, $\overrightarrow{O\text{'}A}=\left\{0,-r_b\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\beta }_A,-r_b\cdot \mathrm{s}\mathrm{i}\mathrm{n}{\beta }_A\right\}F$, $\overrightarrow{O\text{'}B}=\left\{0,r_b\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\beta }_B,r_b\cdot \mathrm{s}\mathrm{i}\mathrm{n}{\beta }_B\right\}F$.

3.3. Contact point movement on nut side

According to the drive principle of ball-screw mechanism, when the displacement of screw is $L$, the nut rotation angular displacement will be $2\pi L/p$, where $p$ represents the screw lead. So, when the velocity of screw is $dL/dt$, the angular nut velocity can be written as:

Then transform it into the Frenet frame:

Therefore, the velocity of contact point $A$ on the screw can be denoted as:

3.4. Contact point movement of on screw side

The velocity of contact point $B$ in the World Coordinate System can be denoted as:

Transform the Eq. (17) into the Frenet frame:

3.5. Relative sliding speed of contact point

For contact point $A$, according to the velocity $V_{A,ball}$ on the screw and the velocity $V_{A,nut}$ on the nut side, the relative sliding speed of contact $A$ can be obtained as:

Similarly, the relative sliding speed of contact $B$ can be obtained as:

As shown in the Fig. 4 above, due to the existence of contact angle in the ball-screw mechanism, while the ball rolls along the curve of the ball center, a spin-slip phenomenon around the common normal of contact point will also simultaneously exist.

The inerter is commonly used in the low and medium frequency workplaces, and the common normal of contact point $A$ and $B$ are both inside the normal plane $b-n$, so it can be approximately deemed as ${\beta }_A={\beta }_B=\beta$. And according to the Plane Vector Geometry: $\overrightarrow{O^{\text{'}}A}//\overrightarrow{O^{\text{'}}B}$, it means the common normal of two contact points are duplicated. Assume the angular velocity when the ball is rotating around common normal as ${\omega }_{o\text{'}}$, and then obtain the following equation set:

Fig. 4Schematic of ball movement

Substitute the equation set Eq. (21) into the expression of $V_A$ and $V_B$, and then convert it into angular velocity rotating around common normal:

4.1. Interface stress

As shown in Fig. 5, according to the Hertz contact theory, the shape of the contact surface between the ball, nut, and screw is ellipse. Assume that the major semi-axis of ellipse between the ball and nut is as $a_A$, and the minor semi-axis of ellipse is as $b_A$. Then, create a Cartesian coordinate system ($O_A$-$X_A,Y_A,Z_A$) by taking the ellipse center as its origin, where $X_A$ coincides with the major axis of the ellipse, $Y_A$ coincides with the minor axis, and $Z_A$ is perpendicular to the $X_A$-$Y_A$ plane; Similarly, the major semi-axis of the ellipse between the ball and screw is assumed as $a_B$, and the minor semi-axis is assumed as $b_B$. Then, create a Cartesian coordinate system ($O_B$-$X_B,Y_B,Z_B$) by taking the ellipse center as its origin $O_B$, where $X_B$ coincides with the major axis of the ellipse, $Y_B$ coincides with the minor axis, and $Z_B$ is perpendicular to the $X_B$-$Y_B$ plane. Due to space limitation, the calculating methods of major semi-axis $a_A$ and $a_B$, minor semi-axis $b_A$ and $b_B$ can be referred to the methods in literature [22].

Fig. 5Schematic of contact area between ball and raceway

Fig. 6Schematic of contact stress distribution

As shown in Fig. 6, take the contact area between the ball and screw as an example, inside the ellipse contact area, the contact stress of every point is different. On axis $Z_B$, due to the maximum deformation, its contact stress reaches the maximum value too. The contact stresses of other points are distributed according to the half ellipsoid rule, and these values are [22]:

where, $q$ represents the stress of random point inside the ellipse contact area, $q_{\mathrm{m}\mathrm{a}\mathrm{x}}$ represents the maximum stress inside the ellipse contact area.

The stress of the ellipse contact area is distributed according to the half ellipsoid rule, so the positive pressure acting on a single working ball, and the stress of every point has an integral ratio as follows [22]:

where, $S$ represents the elliptical contact area.

The Eq. (24) shows that in order to get the stress of random point inside the ellipse contact area, the maximum stress $q_{\mathrm{m}\mathrm{a}\mathrm{x}}$ inside it has to be calculated first. The following shows the solution of maximum stress $q_{\mathrm{m}\mathrm{a}\mathrm{x}}$.

As shown in Fig. 7, due to the existence of lead angle $\alpha$, the axial excitation load $F_{in}$ acting on the single working ball is $F_{in}\cdot \mathrm{c}\mathrm{o}\mathrm{s}\alpha$, so the positive pressure acts on a single working ball can be obtained as:

where $Z$ represents the ball number in the ball-screw mechanism;

Synthesize Eq. (25) and Eq. (26), the maximum stress inside the ellipse contact area:

Fig. 7Schematic of stress analysis

4.2. Friction torque of contact area

As shown in Fig. 9, take the contact between the screw and ball as an example, take a random point $Q(x,y)$ in the coordinate system, and its area element $dS=dxdy$, then its friction can be obtained as $d{F}_f={\mu }_f\cdot q_{Q}dxdy$, where ${\mu }_f$ refers to the friction coefficient of contact area, and $q_Q$ refers to the positive pressure of point $Q$, continue to get its moment of the area relating to axis $Z_B$ as $d{M}_B=\sqrt{x^2+y^2}d{F}_f$ based on the length between $Q$ and $B$.

Fig. 9Schematic of contact surfaces infinitesimal

Thus, the moment of the whole area relating to axis $Z_B$:

Decompose $M_B$ to the three-coordinate axis of Frenet frame:

where $M_{BF}$ represents the friction torque of contact point $B$ in the Frenet frame.

4.3. Sliding friction efficiency

The angular velocity and friction moment of two contact points $A$ and $B$ in the Frenet frame have been obtained above, so the its friction power is respectively shown as bellow:

where $M_{AF}$ represents the friction moment of contact point in the Frenet frame;

The sliding friction power of single ball is a sum of two equations above:

Since the physical quantity power is a scalar, its value will not change along with the change of coordinate system, thus the excitation power while in the Frenet frame can be expressed as:

Define the energy consumption of rolling friction that account for the percentages of excitation as sliding friction efficiency:

4.4. Rolling friction efficiency

Define the energy consumption of rolling friction that account for the percentages of excitation as rolling friction efficiency [23]:

where ${\mu }_f^{\text{'}}$ represents the rolling friction coefficient, generally within the range of 0.0036-0.0038.

6.1. Experimental verification

To verify the derived nonlinear inertance and the accuracy of this calculation method, this research has performed a testing experiment to the inertance of the ball-screw inerter on the ES-10-240 vibration test bench, the parameters of ball-screw mechanism are shown in Table 1 and the setup of vibration test bench is shown in Fig. 10. This experiment has also installed a force sensor at the connection between the ball-screw inerter and the vibration test bench, and installed several vibration acceleration sensors on the vibration table at the same time, and then the DASP V10 model analyzer was used for simultaneous vibration data acquisition and signal spectrum analysis, by loading different frequencies of square wave excitation to the ball screw inerter, which are within the frequency range from 1 Hz to 12 Hz, for every rises of 1 Hz per test, and the experimental results of force and acceleration are recorded in Table 2.

Fig. 10Experimental device and its live photos

a) Experimental device

b) Living photos

Table 2 is organized into Fig. 11, with the frequency rises, both ends of the force and acceleration have the increasing trend, according to the definition of inertance: $\mathrm{\Delta }F=b\cdot \mathrm{\Delta }a$, where $\mathrm{\Delta }F$ is the force of two ends, $\mathrm{\Delta }a$ is the acceleration of two ends, the experimental results of inertance can be obtained, as shown in Table 3.

After that, fit the test points by using MTALAB cftool tool box. In addition, the experimental fitting curve is compared with the nonlinear inertance and the ideal linear inertance, thus obtaining the following graph Fig. 12.

These experimental results show that the nonlinear inertance derived by this paper is slightly larger than the experimental fitting results. When the frequency is less than 12 Hz, the error between the ideal linear inertance and the experimental inertance is greater than 10.0 %. It can be known that the ideal linear inertance is not accurate enough in a low frequency range of the ball-screw inerter, in some demanding vibration isolation systems; the use of the ideal linear inertance will introduce large errors. However, when the frequency is less than 12 Hz, the error between the nonlinear inertance and the experimental inertance is less than 5.0 %, especially when the frequency is greater than 3 Hz, the error between the nonlinear inertertace and experimental inertance is less than 2.0 %. Obviously, compare with the ideal linear inertance of the ball-screw inerter, the nonlinear inertance derived in this paper is more accurate in the low frequency range of the inerter.

Fig. 11Experimental test force and acceleration

a) Force

b) Acceleration

Fig. 12Inertance obtained by different methods

6.2. Parameter evaluation

Fig. 13 shows the comparison of the influence of different lead on the nonlinear inertance, the other parameters are shown in Table 1. At the same frequency, the smaller is the lead, the greater is the inertance, the stronger is the nonlinearity of inertance of the ball-screw inerter. Additionally, as the frequency increases, the nonlinear inertance tends to be stable.

Fig. 13Influence of different lead on inertance

Fig. 14-Fig. 17 show the comparison of the effects of nominal radius, ball radius, ball number, and contact angle on the nonlinear inertance respectively, the other parameters are shown in Table 1. At the same frequency, the larger is the nominal screw radius, the smaller is the ball radius, the larger is the number of balls and the larger is the contact angle, so the nonlinear inertance will be smaller, and the nonlinearity of the inertance will be stronger. In addition, within 3 Hz, the frequency has a great influence on the nonlinearity of inertance, the smaller is the frequency, the stronger will be the nonlinearity of inertance and the smaller will be the inertance; As the frequency increases, the nonlinear influence of the frequency on the inertance is weakened, and the inertance tends to be stable.

Fig. 14Comparison of influence of different nominal radius on inertance

Fig. 15Comparison of influence of different ball radius on inertance

Fig. 16Comparison of influence of different ball numbers on inertance

Fig. 17Comparison of influence of different contact angles on inertance

This paper only considers the influence of the parameters of the ball-screw mechanism in the inerter, and the coupling between the friction coefficient and the friction torque is not taken into account. [24] shows that the change in friction torque will change the viscous resistance of the lubrication medium to which the ball is subjected, which will result in a different friction coefficient. With the increase of the frequency, the difference in the friction coefficient will result in the nonlinear inertance close to the ideal linear inertance in a fluctuating shape.