Experimental investigation and modeling of nonlinear, adaptive dashpot

In this paper we determine the characteristics of mechanical components of dynamical systems using a specially designed laboratory rig. We present the details of the experiments performed on the prototype device and provide technical documentation of its crucial elements. We validate the accuracy of measurements using a spring and comparing results with manufacturer data. Then, we examine nonlinear dashpot with variable damping coefficient obtained through the usage of the throttling valve. We perform several tests presenting its force characteristics as a function of velocity and damping coefficient. We derive the mathematical model of the dashpot basing on the experimental data. Finally, we perform transient test in which we change the damping coefficient during operation of the dashpot. The comparison of obtained results with the model gives good accordance.


Introduction
The engineering constructions are often subjected to unwanted vibrations that in some cases can be mitigated by means of a viscous damper. It is a device that dissipates the excess of the kinetic energy thanks to the viscosity of the uid moving inside. As the layers of a uid move against each other, the kinetic energy is changed into heat and dissipated to the environment [1,2]. Dampers of this kind are widely used in civil engineering [3,4], aerospace [5,6,7] and automobile [8,9] industries. In the last listed eld,a viscous damper is often called dashpot and is the most frequently used as a part of the vehicle suspension. We can distinguish the linear and rotary dashpots, with respect to the manner in which they perform motion. Nevertheless, direction of motion should not be confused with velocity-force characteristic which can be either linear or non-linear in both types of dashpot. The simplest devices are designed to be linear what corresponds with the assumption of the purely linear characteristics hence the force generated in the dashpot is proportional to relative velocity of device terminals. The more precise model takes into account that the force is proportional to the relative velocity raised to required power, that depends on a particular dashpot construction [3,10,11,12]. The damping force depends on the geometry of the dashpot and of the uid used. Both factors are optimized to suite the working conditions of the device such as the frequency and the power of damped oscillations. In some applications the working conditions of the damped system change over time and there is a need to vary the characteristics of the damper. Such a device is called the semi-active damper and it is used in many industrial applications. For example, one could expect a car suspension to behave dierently for varying road conditions [13,14,15]. A controller connected to a system of sensors monitors the working conditions and adjusts the damping properties to the current needs. Dampers with variable damping coecients are also widely used in mechanical and structural systems for seismic and wind storms protection of high buildings [16,17,18], control of vibrations of wind turbines [19,20,21,22], stabilizing washing machines [23], chatter suppression [24,25], high impulsive loads [26,27,28], stabilization of exible structures [29] and many more. One can also nd applications in medicine where the damper is a part of prosthesis [30,31]. As the damping depends on the geometry of the dashpot and on the uid viscosity, the adjustment can be done in two ways. First of the popular solutions relies on the properties of the magnetorheological uid [32]. Its viscosity can be controlled by changing the magnetic eld around [33,7], typically by changing the current in electromagnets. The other solution is based on the change of geometry of the ow. An adjustable throttling valve locally changes the cross section of the ow [34,35] restraining or relaxing the motion of the uid.
In this article a semi-active dashpot with a throttling valve is experimentally analyzed. We design the dedicated rig that enables to measure the acting force of damper at dierent relative velocities of its nodes and at dierent positions of the throttling valve. We show the mathematical model of investigated device that match perfectly to experimental data. Presented results are a part of the on going work on the properties of the tuned mass damper with the adjustable dashpot [36,37,38].
The paper is organized as follows. In Section 2 we describe the test stand. Section 3 is devoted to validation of correctness of all components of the experimental rig. We describe measurements of the stiness of the spring and extensive experimental tests of the damper. In Section 4 we create a mathematical model of the dashpot basing on previously obtained experimental results. Then, in Section 5 we validate the model experimentally simulating the continuous changes of damping coecient. The results are concluded in the last section.

Description of the test stand
The laboratory rig (  . The encoder is a part of the driver's closed loop system but is also used as the main position sensor. An additional linear position sensor (part No. 7) is mounted in parallel to the actuator. It is a linear potentiometer KTC-375 whose resistance is 4 [kΩ]. Measurement of the force generated by the examined devices is possible due to a force sensor (part No. 8) installed at the end of the piston rod. The universal mounting bracket (part No. 9) between the sensor and the piston rod is designed to ensure a possibility of using dierent models and types of sensors. The same mounting bracket is used to fasten the piston rod of the potentiometer. For the experiment we use an s-type load cell m620 nominally measuring the forces up to 500 [N], that can be safely overloaded to 1000 [N]. In order to protect the cable of the sensor, we put it in a plastic cable chain Igus R07 (part No. 10). The second end of the force sensor is connected to the dashpot through a combination of spherical and cylindrical joints (part No. 11). This type of connection reduces the minimal assembly misalignment. The connection can also be easily used to mount various types of devices such as springs, dashpots and inerters. In this particular experiment we examine the semi-active dashpot Ohlins SD 043 (part No. 12). The stroke of the dashpot is equal to 60 [mm]. It has an adjustable damping coecient, regulated by a built in stepper motor. The motor drives a throttling valve, which changes the cross section of the channel through which the damping liquid is owing. The dashpot is mounted in a specially designed bracket (part No. 13) xed to a platform similar to the one put under the actuator (part No. 14).
The test rig is controlled through a PC connected to the driver of the servomotor via the USB interface. By means of the dedicated software -Mitsubishi MR Congurator2 -the position, the velocity and the acceleration of the motor in time can be precisely adjusted. The motor we use has a power reserve so in case of the software failure it could easily destroy the other components of the rig, therefore we applied the following preventive measures. Firstly, the torque of the motor is software-limited to 20 % of the nominal value resulting in 3.4 [Nm], which corresponds to 1000 [N] of the axial thrust. This is the maximal load that the force sensor can stand. Secondly, close to the piston rod stroke limits, there are two magnetic eld proximity sensors SMT 65 (part No. 14), which act as safety switches. When the piston rod exceeds the limits imposed by the sensors they stop the servomotor. Finally, we placed an adjustable mechanical brake (part No. 15), on which the piston rod of the actuator will stop if both the software and the switches fail. A position of the brake can be adjusted along the aluminum prole Item 8 40x40 (part No. 16) according to the accessible stroke of the object under investigation. Also for the safety reasons we limit the acceleration of the drive.
The damping of the dashpot is also regulated from the PC, through an own software written in Ansi C.

Experimental results
In the rst part of this section, we show experimental validation of the test rig. While in the second part, we analyze semi-active damper with variable damping coecient presenting several velocity-force characteristics for dierent values of damping coecient.

Validation of the test rig
To validate the correct assembly and operation of the test rig we perform series of tests. In all of them the device has been equipped with three 3-axial accelerometers (one mounted on the end of the piston of the linear actuator, one on the dashpot and one on the rig) to measure oscillations of the system during operation. Based on accelerograms we are able to minimize vibrations in directions perpendicular to the axis of motion by slight corrections of the assembly of all elements in measurement chain and ensure their coaxiality. The next stage of test has been focused on comparison between assumed velocity and acceleration proles assumed in control software and real motion of the device.
In the very last test we measure the characteristic of a linear helical extension spring of a known stiness. According to the supplier documentation the spring used in the experiment has the stiness of 3. 29 .125.180.0630.A). The spring is preloaded with the force of 10 [N] in order to overcome the initial tension of the spring and to remove the play present in the system. Next, it is elongated by 40 [mm] and obtain force -displacement relation. The stiness of the spring is derived from the linear t obtained by the method of least squares. The measurement has been repeated ten times and the coecient of determination R 2 of each t has been above 0.99. The mean stiness coecient is equal to 3.29 N mm . To be sure that there is no relation between the measured force and the velocity of motion the spring has been also elongated at dierent velocities of the actuator and indeed no relation has been found.   Fig. 3(a) for the maximum velocity equal to 0.066 m s . Fig. 3(b) shows the force measured at this velocity for the control parameter set to γ = 100 [−]. It is worth to notice, that the force in Fig. 3(b) starts increasing before the motion starts. This eect comes from the dry friction present in the dashpot. We assume, that the velocity is constant when its value is within the range v ± 1%v, where v is the expected value of constant velocity. Then, the force F is measured in the whole range of constant velocity and as a result we take the mean value of force. Such a value corresponds to one measurement point presented in Fig. 4, which shows a summary of all measurements taken in the experiment. The measurement points are marked as dots while the lines are the linear interpolations between the points which are taken for the same value of damping coecient.

Model of the dashpot
The measurements performed for the investigated dashpot are used to create its mathematical model. The force F generated by the device has been measured with respect to two variables, absolute velocity v and control parameter γ. Hence, we are able to create a two parameter surface given by a function F = f (v, γ). We t a polynomial surface to the data points by the RMS method. The order of polynomial is equal to three and ve in v and γ directions respectively. It is the lowest order of polynomial, that reected well all the changes of the measured force and it results in very good tting given by R 2 = 0.999. The obtained formula is as follows:  Fig. 5. To present the changes in the force with increase of velocity and control parameter γ we use a color scale for surface, additionally by blue bullets we mark the experimentally measured values of force.
In order to have a better view on the response of the dashpot on changes of two control parameters, we plot two projections of the surface in Fig. 6. In the rst projection presented in Fig. 6(a) we plot 14 force -velocity characteristics for γ ∈ 0, 130 [−] with 10 [−] units step. The mathematical model ts well with measured data (marked by dots). Results visible in this plot has been described in the previous section. The results shown in Fig. 6(b) correspond to projection of surface on forceγ plane, hence plotted lines refer to constant velocity. We can observe that for small velocities the inuence of γ parameter is not very signicant but with an increase of velocity the inuence become noteworthy and strongly nonlinear with hardening character. We assume that the origin of the non-linearities lies in the properties of the uid inside the dashpot and its ow through a throttling valve of a variable cross-section. A detailed description of the dashpot from the viewpoint of uid mechanics is beyond the scope of this work. As it is easy to see, the matching of data is very good and we claim that the presented model can be used in to simulate the behavior of the dashpot in real life implementations.. . Finally, we decelerate the rod to the full stop. The change of the dashpot control parameter is logged on the data acquisition system in parallel with the force. By combining two curves with respect to the variable dashpot control parameter γ, we obtained results presented in Fig. 7. We show the comparison of the experimental result (blue continuous line) and the response calculated based on mathematical model (red continuous line). On both sides of the numerical trace of the force we the plot dash-dot lines which show the 95% condence bound for future observations. The experimental results do not cross any condence bound lines. Hence, we claim that the matching of experimental and numerical data is very good. Thus, we eects of the stepper motor control are minor and the investigated dashpot can be modeled as we propose in Section 4 also for applications with continuous change of controlling parameter γ.

Inuence of the stepper motor control
Another important factor is the possible rate of change of the damping coecient. In the considered device the change of control parameter γ from its minimum to maximum value takes at least 0.34 [s].In our test we executed the change from minimum to maximum value in 0.44 [s] which refers to 77% of maximum rate of change and can be considered as fast change in damping coecient. Although the stepper motor enables fast changes without aecting the dynamics of the system one has to bear in mind the maximum possible rate of change of damping coecient.

Conclusions
In this paper we show the comprehensive experimental investigation of the semi-active dashpot. In order to examine its characteristics we create a specially designed laboratory rig. Using a simple helical spring we   legitimize the concept of the rig and prove that it enables high precision measurements to examine elementary mechanical components of dynamical systems like springs, dampers and inerters Then, we preform comprehensive experimental test of the semi-active dashpot with a throttling valve Ohlins SD 043. Based on the obtained experimental data we develop a mathematical model of the device. Its parameters are identied using two dimensional surface t implemented in Matlab. We present the force-velocity and force-control parameter γ dependence graphs combined with experimental data. In both cases we obtain very good matching. We observe that the dashpot characteristic is strongly non-linear of hardening type. Hence, for low velocities and small values of γ it is close to linear, but with the increase of those parameters it becomes signicantly steeper and clearly diverges from linear characteristic. We also investigate the eects of noncontinuous changes of damping coecient performed by a stepper motor. The results show that the investigated device is capable of fast changes of damping coecient and without disturbing smooth operation. This proves the robustness of the device and the proposed mathematical model. According to the good agreement between the experimental and simulation results we claim that our model is accurate enough to be used in simulations of dynamical systems equipped with controllable dashpot with throttling valve.