1 Introduction

The motivation for solving the problem of thin, flexible structure vibration is related to the industrial needs. The achievable positioning accuracy of the end point under dynamic load is crucial for these structures. A typical example of such structure is machine tool quill; it has a thin shape with limited stiffness. This problem can be solved using the auxiliary cable structures, which are not difficult to attach to the specific point of the flexible structure. The desired motion of whole system can be achieved by appropriate control actions at the cable structure.

For undesired vibration suppression, dampers are often used, either a passive damper tuned to a specific frequency [1, 2] or an active damper which is able to influence a wide band of frequencies. Publication [3] describes various options for dealing with vibration, such as semi-active vibration insulation or active vibration damper for machine tools. Work [4] deals with the design of a 3-degrees-of-freedom active vibration absorber for a robot arm. A special case of an active damper with delayed feedback, called delayed resonator, which is able to perfectly suppress vibration of the primary structure, has been studied extensively [5, 6], also with acceleration feedback [7] or combination of position, velocity, and acceleration feedback [8]. Suppression of multiple frequencies is investigated in [9] and [10]. In [11, 12], the concept has been extended for planar vibration suppression. The underactuated mechanical systems’ residual vibration cancelation is investigated in [13, 14]. The papers show the wide range of vibration suppression methods, which often need significant mechanical modification of the machine working point (TCP).

In manufacturing machines, the vibrations can be dealt with by modifying the work-piece holder [15], the tool holder [16, 17], or the tool itself [18]. In [19], a composite boring bar is developed to dampen vibrations of the tool and in [20] a variable stiffness vibration absorber is put inside the tool. The vibration can also be reduced by modifying the fixture, for example partly submerging the work-piece in magnetorheological fluid [21]. In [22], vibration of the work-piece holder is dampened using a magnetorheological damper. However, using existing actuators with a new control strategy allows to suppress vibrations without adding additional mass to any part of the machine [23]. Another solution is proposed in [24] combining input shaping with the addition of servo control for uncertainties and disturbances removing. In [25], the prediction of the position error is used to modify the input command of a dual-driving gantry-type machine to improve accuracy. In [26], a control strategy is designed to reduce vibrations of the tip of a long and flexible manipulator. These papers propose direct placement of additional actuators, change of materials, or complicated control strategies. Altogether these ways need fundamental changes of the machine structure or machine control concept.

Actuators in the form of piezoelectric patches have been used to reduce the vibration of flexible beams in [27] and [28] using a delay-feedback controller, and in [29] using a self-organizing map-based controller. Work [30] studies the dynamics of a cantilever stayed beam. The dynamic properties of a cable-beam system with cantilever beam and the wires connected to its tip have been explored with wires being perpendicular in [31] and parallel [32]. A flexible manipulator with two cables is studied in [33]. Publication [34] deals with vibration of a tapered cantilever beam with a wire connected to its tip. The wires, in combination with a control strategy, can be used to suppress vibrations of the beam, such as in [35]. To be able to transfer force through a cable, the dynamical properties of the cable must be also known and included in the control. The behaviour of a fibre has been modelled with respect to fibre mass, stiffness, and damping [36,37,38,39]. In [40], stiffness of cables with large strains has been studied. These papers deal with the cable properties, appropriate models, or control without machine tool context.

An approach to studying the position accuracy and vibrations of a high-frequency moving robot arm has been proposed in [41]. In [42], vibrations of cable mechanisms and their properties in general have been studied and an optimal Fuzzy-PID control developed, focusing on optimizing settling time, maximum control force, and energy consumption. In [43], an adaptive robust control is developed for cable-driven parallel robots for trajectory tracking. The machine redundancy is investigated in [44] and control with sliding mode control is presented in [45]. Control of mechanical systems with particular cable application is shown in the papers.

The presented paper brings a new idea of the vibration suppression of production machine working point (TCP). Several types of machine tools with a large workspace suffer from unwanted vibrations of long and heavy movable parts. These vibrations are very often composed mainly from bending vibration with first eigenfrequency of such parts and their additional suppression can improve material removal abilities and spindle power utilization in the whole working span. The novelty of concept is based on adding cables controlled by standard rotational drives, which create the auxiliary parallel structure and bring additional damping force near to the machine working point (TCP). According to the best knowledge of the authors, the investigation of the machine tools cable-based vibration suppression is so far very limited. Moreover, the cables can be easily reconfigured and the original workspace of machine is almost preserved. The simplified model of machine tool structure is used, the model is prepared, and the control law is established. The simulation experiment with state feedback is performed as well as experimental work.

This concept can help many types of machines with an upper gantry and a slim headstock box-in-box positioned structure, e.g. machines from the TAJMAC-ZPS portfolio. This approach leads to increased material removal capabilities and the use of spindle power over the entire working range, or allows the design of the machine tool to be lightened.

The paper is organized as follows. In the first section, the motivation and literature survey is prepared, the second section shows the experimental setup model with identification of all important parts, and the system is linearized. The third section is dedicated to the vibration damping analysis with two ways of drive control and also shows the simulation experiment of the resulting structure. The fourth section is dedicated to experimental results. The last section concludes the work.

2 Experimental setup modelling

A physical model of experimental setup is derived in this section. To explore the concept of suppressing vibrations using additional cables, simple experimental demonstrator was prepared as shown in Fig. 1. The demonstrator is created in order to include the main dynamic properties of the abovementioned machine tools with a large workspace and long heavy movable parts.

Fig. 1
figure 1

Experimental stand

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Fig. 2
figure 2

Mechanical model of the experimental stand

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Based on the stand, a simplified simulation model was created; Fig. 2 shows model parameters and coordinates. The torques \(M_i\) are exerted by electric servo drives with inputs \(u_i\). The cart movement is prescribed as separate input, and the cable mass is neglected. A free body diagram method is chosen to assemble the equations of motion. Forces and torques acting on each body of the system and coordinates describing their movement are depicted in Fig. 3. Equations of motion resulting from Fig. 3 are

$$\begin{aligned} \begin{aligned} I_1\ddot{\varphi }_1&= M_1 - F_1r_1 - b_{M1}\dot{\varphi }_1 \\ I_2\ddot{\varphi }_2&= M_2 + F_2r_2 - b_{M2}\dot{\varphi }_2 \\ m\ddot{x}_n&= -F +F_2\cos {\alpha _2} -F_1\cos {\alpha _1} \end{aligned} \end{aligned}$$
(1)

where coordinates \(\varphi _1\) and \(\varphi _2\) are electric servo drive rotor angular positions, coordinate \(x_n\) is horizontal coordinate of the TCP, and x coordinate represents horizontal position of the cart.

Fig. 3
figure 3

Free body diagram

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Equations of motion (1) contain inertial forces, cable forces, elastic beam force, damping forces with damping coefficients \(b_{M1}\) and \(b_{M2}\), and cable angles \(\alpha _i\). Forces F, \(F_1\) and \(F_2\) can be expressed as

$$\begin{aligned} \begin{aligned} F_1&= \frac{k_1}{L_1}\xi _1+\frac{b_1}{L_1}\dot{\xi }_1 \\ F_2&= \frac{k_2}{L_2}\xi _2+\frac{b_2}{L_2}\dot{\xi }_2 \\ F&= k(x_n-x) + b(\dot{x}_n-\dot{x}) \end{aligned} \end{aligned}$$
(2)

Determination of cable extensions \(\xi _i\) in Eq. (2) is based on Fig. 4. The left part of the picture depicts the derivation of angle \(\alpha _1\) measured from the x axis, and the right part shows length of cable including segments wound on pulleys. Stiffness of both cables is presumed to be the same and equals to \(k_L\), \(k_1 = k_2 = k_L\). Stiffness is in reciprocal proportion to the cable length as well as the cable dampings \(b_1\) and \(b_2\), \(b_1=b_2=b_L\).

Fig. 4
figure 4

Derivation of cable angle (left) and cable length (right)

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Extension of cables (longitudinal deformation) equals to difference of actual cable length (including small cable segments on the pulleys, see Fig. 4 right) and initial length of the cable:

$$\begin{aligned} \begin{aligned} \xi _1&= L_1 -L_{10} + (r-r_1)(\alpha _1-\alpha _{10}) + \varphi _1 r_1 \\ \xi _2&= L_2 -L_{20} + (r-r_2)(\alpha _2-\alpha _{20}) - \varphi _2 r_2 \\ \end{aligned} \end{aligned}$$
(3)

Symbols \(L_{10}\), \(L_{20}\) and \(\alpha _{10}\), \(\alpha _{20}\) in Eq. (3) correspond to initial cable lengths and cable angles, respectively. Actual lengths and angles of the cables can be derived from Fig. 4 as

$$\begin{aligned} \begin{aligned} L_1&= \sqrt{{(x_n+\Delta x_1)}^2 + {(y_n-y_1)}^2 - {(r_1-r)}^2} \\ L_2&= \sqrt{{(\Delta x_2-x_n)}^2 + {(y_n-y_2)}^2 - {(r_2-r)}^2} \\ \alpha _1&= \beta + \gamma = \arctan {\left( \frac{y_n-y_1}{x_n+\Delta x_1}\right) } + \arctan {\left( \frac{r_1-r}{L_1}\right) } \\ \alpha _2&= \arctan {\left( \frac{y_n-y_2}{\Delta x_2-x_n}\right) } + \arctan {\left( \frac{r_2-r}{L_2}\right) } \end{aligned} \end{aligned}$$
(4)

with the only time-dependent variable \(x_n\), the rest of the parameters are constants of the structure design. Radii \(r_1\) and \(r_2\) are first and second motor cable pulleys, and r is the radius of beam tip cable pulley, where the cables are fixed. The beam deformation is assumed to be small; therefore, the beam tip coordinate \(y_n\) does not change its value.

Cable extension velocities \(\dot{\xi }_1\) and \(\dot{\xi }_2\) used in Eq. (2) are time derivative of Eq. (3) and particular members of Eq. (4):

$$\begin{aligned} \begin{aligned} \dot{\xi }_1&= \dot{L}_1 + (r-r_1)\dot{\alpha }_1 + \dot{\varphi }_1 r_1 \\ \dot{\xi }_2&= \dot{L}_2 + (r-r_2)\dot{\alpha }_2 - \dot{\varphi }_2 r_2 \\ \end{aligned} \end{aligned}$$
(5)

The electromagnetic field in the BLDC drives produce the rotor torques \(M_1\) and \(M_2\). The torque is modelled as proportional to the stator armature current:

$$\begin{aligned} \begin{aligned} M_1&= k_{t1}i_1 \\ M_2&= k_{t2}i_2 \\ \end{aligned} \end{aligned}$$
(6)

with the current constants \(k_{t1}\) and \(k_{t2}\) of the first and second motor.

The moments of inertia \(I_1\) and \(I_2\), the equivalent beam mass m, damping coefficients \(b_{M1}\) and \(b_{M2}\) from the equations of motion in Eq. (1), stiffnesses k and \(k_L\), damping coefficients b and \(b_L\) from the forces in Eq. (2), and parameters \(k_{t1}\) and \(k_{t2}\) from (6) are subject of identification process while being unknown.

2.1 System parameters identification

Model parameters are assessed in this sub-section. The important property of all parameters lies in the linearity. They create a system of linear equations (mostly time dependent). Due to the measurement of other system parameters than just inputs and output, the identification process is separated into three parts: beam identification, cable properties identification, and drive 1, drive 2 identification.

The beam properties are obtained from the measurement of initially deflected beam. An accelerometer is placed at the end point of the beam and free oscillations are performed due to the initial beam deflection. The measurement is performed by an accelerometer and oscilloscope. For the cable and electric drives with drive controller part identification is performed following experiment. The sweep cosine signal is an input at first drive and constant pre-load is applied at the second drive. The similar experiment is performed with constant pre-load at first drive and sweep cosine signal at the second drive. The excitation frequency range is from the 0.5 up to 30Hz, which is a reasonable range for expected machine tool movement excitation. The horizontal position of the beam end-point and cart position is measured by laser-tracker, and the drive angular positions are measured by drive internal measurement sensors.

2.1.1 Beam properties

The beam fixed to the cart is modelled as a cantilever and it dominantly vibrates at its first eigenmode. The identification of k, b, and m is based on analytical solution of beam deflection and experimental data.

Fig. 5
figure 5

Identification of beam parameters

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The experiment on Fig. 5 shows unforced beam oscillations measured by an accelerometer. On the left part of the figure, two beam oscillation periods are measured and damped system eigenfrequency is obtained in Eq. (7).

$$\begin{aligned} \begin{aligned} \Delta t&= 0.185~s \\ f_{beam_d}&= \frac{2}{\Delta t} = 10.8108~Hz\\ \end{aligned} \end{aligned}$$
(7)

Damping ratio is obtained from Fig. 5 right part. The mean accelerometer output signal value is \(Signal_{mean}~=~4.42mV\), the peak value at the oscillation beginning is \(Signal_1~=~608mV\), and the oscillation amplitude after \(\Delta t_{sig}~=~6.28s\) is \(Signal_2~=~88mV\). Logarithmic decrement \(\delta\) is natural logarithm of magnitudes ratio:

$$\begin{aligned} \begin{aligned} \delta&= \ln {\frac{Signal_1-Signal_{mean}}{Signal_2-Signal_{mean}}} = 1.977 \\ \delta&= 2\pi f_{beam_d} \Delta t_{sig} \kappa \\ \end{aligned} \end{aligned}$$
(8)

with \(\kappa\) used in the damping ratio \(\zeta\) evaluation:

$$\begin{aligned} \begin{aligned} \kappa&= \frac{\delta }{2\pi f_{beam_d} \Delta t_{sig}} \\ \zeta&= \frac{\kappa }{\sqrt{\kappa ^2+1}} = 4.635\cdot 10^{-3} \\ \end{aligned} \end{aligned}$$
(9)

Using damped system eigenfrequency in Eq. (7) and damping ratio in Eq. (9), the undamped beam eigenfrequency (further just eigenfrequency) is obtained:

$$\begin{aligned} \begin{aligned} f_{beam}&= \frac{f_{beam_d}}{\sqrt{1-\zeta ^2}} = 10.8109~Hz\\ \end{aligned} \end{aligned}$$
(10)

Damping ratio and eigenfrequency are not sufficient to obtain k, b, and m parameters of the reduced beam model. The equation of fixed beam horizontal deflection allows us to calculate beam horizontal stiffness at beam reduced mass position. Beam length \(L~=~0.95m\), material young modulus \(E~=~70\cdot 10^9Pa\), and cross-section quadratic moment \(J~=~1.88\cdot 10^{-8}m^4\) are required inputs for beam stiffness calculation:

$$\begin{aligned} \begin{aligned} k&= \frac{3EJ}{L^3} = 4604.75 \frac{N}{m}\\ \end{aligned} \end{aligned}$$
(11)

Then using homogeneous part comparison of the third equation of motion from Eq. (1) with substituted force F from Eq. (2)

$$\begin{aligned} \begin{aligned}&\ddot{x}_n+&\frac{b}{m}\dot{x}_n&+&\frac{k}{m}x_n = Rhs\\&\ddot{x}_n+&2\zeta (2\pi f_{beam})\dot{x}_n&+&(2\pi f_{beam})^2x_n = Rhs\\ \end{aligned} \end{aligned}$$
(12)

beam equivalent mass m and beam damping b are derived:

$$\begin{aligned} \begin{aligned} m&= \frac{k}{(2\pi f_{beam})^2} = 0.998~kg\\ b&= 2\zeta (2\pi f_{beam})m = 0.63~\frac{Ns}{m}\\ \end{aligned} \end{aligned}$$
(13)

2.1.2 Cable properties

The cable used at experimental stand at Fig. 1 is a lightweight wound metallic cable characterized by its stiffness \(k_L\) and damping \(b_L\). The cable is the same on both sides of the experimental setup. The cable properties identification is based on the experimental measurement. Sweep cosine function is introduced at drives with prestress preventing the cable unloading. The inputs (angular positions \(\varphi _1\) and \(\varphi _2\)) and the horizontal position of beam TCP point \(x_n\) as well as cart position x are recorded.

The equation of motion, taken into account for the identification analysis, is the third one of the set of Eq. (1). Introducing the forces Eq. (2) into this particular equation, it is obtained:

$$\begin{aligned} \begin{aligned}&m\ddot{x}_n+b\left( \dot{x}_n-\dot{x}\right) + k\left( x_n-x\right) =&\\&=k_L\left( \frac{\xi _2}{L_2}\cos {\alpha _2}-\frac{\xi _1}{L_1}\cos {\alpha _1}\right) +b_L\left( \frac{\dot{\xi }_2}{L_2}\cos {\alpha _2}-\frac{\dot{\xi }_1}{L_1}\cos {\alpha _1}\right) \\ \end{aligned} \end{aligned}$$
(14)

where unknown parameters \(k_L\) and \(b_L\) are in linear combination. Other parameters and terms are known or could be derived from the measured data and its numeric time derivative. Numeric time derivative is amended with non-causal zero-phase low pass data filtering.

Data acquisition and further processing keeps the measurement sampling frequency \(f_s = 1kHz\). The Eq. (14) is expanded in time and the following matrices are created:

$$\begin{aligned} \mathbb {U} = \begin{bmatrix} m\ddot{x}_n(0)+b\left( \dot{x}_n(0)-\dot{x}(0)\right) + k\left( x_n(0)-x(0)\right) \\ m\ddot{x}_n(\Delta t)+b\left( \dot{x}_n(\Delta t)-\dot{x}(\Delta t)\right) + k\left( x_n(\Delta t)-x(\Delta t)\right) \\ \vdots \\ m\ddot{x}_n(n\Delta t)+b\left( \dot{x}_n(n\Delta t)-\dot{x}(n\Delta t)\right) + k\left( x_n(n\Delta t)-x(n\Delta t)\right) \\ \end{bmatrix} \end{aligned}$$
(15)
$$\begin{aligned} \begin{aligned}&\mathbb {V}_1 = \begin{bmatrix} \frac{\xi _2(0)}{L_2(0)}\cos {\alpha _2(0)}-\frac{\xi _1(0)}{L_1(0)}\cos {\alpha _1(0)}\\ \frac{\xi _2(\Delta t)}{L_2(\Delta t)}\cos {\alpha _2(\Delta t)}-\frac{\xi _1(\Delta t)}{L_1(\Delta t)}\cos {\alpha _1(\Delta t)}\\ \vdots \\ \frac{\xi _2(n\Delta t)}{L_2(n\Delta t)}\cos {\alpha _2(n\Delta t)}-\frac{\xi _1(n\Delta t)}{L_1(n\Delta t)}\cos {\alpha _1(n\Delta t)}\\ \end{bmatrix}\\&\mathbb {V}_2 = \begin{bmatrix} \frac{\dot{\xi }_2(0)}{L_2(0)}\cos {\alpha _2(0)}-\frac{\dot{\xi }_1(0)}{L_1(0)}\cos {\alpha _1(0)}\\ \frac{\dot{\xi }_2(\Delta t)}{L_2(\Delta t)}\cos {\alpha _2(\Delta t)}-\frac{\dot{\xi }_1(\Delta t)}{L_1(\Delta t)}\cos {\alpha _1(\Delta t)}\\ \vdots \\ \frac{\dot{\xi }_2(n\Delta t)}{L_2(n\Delta t)}\cos {\alpha _2(n\Delta t)}-\frac{\dot{\xi }_1(n\Delta t)}{L_1(n\Delta t)}\cos {\alpha _1(n\Delta t)}\\ \end{bmatrix}\\ \end{aligned} \end{aligned}$$
(16)

with \(\Delta t = \frac{1}{f_s}\) and \(n+1\) measured points. The overall set created from the measurement data has unknown parameters formed in the vector:

$$\begin{aligned} \begin{aligned} \mathbb {U}=\begin{bmatrix} \mathbb {V}_1&\mathbb {V}_2 \end{bmatrix} \begin{bmatrix} k_L\\ b_L\\ \end{bmatrix}\\ \end{aligned} \end{aligned}$$
(17)

Unknowns \(k_L\) and \(b_L\) cannot be evaluated directly (number of measurement points is significantly higher than number of unknowns), but they are obtained using least square method leading to pseudo inversion assembly.

$$\begin{aligned} \begin{aligned} \begin{bmatrix} k_L\\ b_L\\ \end{bmatrix} = \left( \begin{bmatrix} \mathbb {V}_1&\mathbb {V}_2 \end{bmatrix}^T\begin{bmatrix} \mathbb {V}_1&\mathbb {V}_2 \end{bmatrix}\right) ^{-1}\begin{bmatrix} \mathbb {V}_1&\mathbb {V}_2 \end{bmatrix}^T\mathbb {U}\\ \end{aligned} \end{aligned}$$
(18)

Resulting stiffness \(k_L = 16432\frac{Nm}{m}\) and damping \(b_L = 131\frac{Nms}{m}\) are verified in the simulation experiment, where the Eq. (14) has angles \(\varphi _1\), \(\varphi _2\) and cart position x as inputs. Comparisons of simulated beam position with experimental beam position measurement are at Figs. 6 and 7. The figures show the time behaviour of beam TCP position (\(x_n\)): the experimental data, simulation data, and their difference as absolute error. Figure 6 presents data based on the engine 1 excitation, and Fig. 7 presents the data from engine 2 excitation. System drift at low frequencies (less than 0.3Hz) is subtracted.

Fig. 6
figure 6

Verification of identified model: cable parameters, motor 1 excitation

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Fig. 7
figure 7

Verification of identified model: cable parameters, motor 2 excitation

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The comparison of model simulation and experimental behaviour in frequency domain is in Fig. 8. The frequency content is similar, the magnitudes slightly differs.

Fig. 8
figure 8

Verification of identified model: cable parameters, frequency analysis

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2.1.3 Electric drives properties

The identification of electric drives parameters is based on the similar experiment as in Sect. 2.1.2. Drive mechanical parameters are according to Fig. 2 rotor moments of inertia \(I_1\), \(I_2\), bearing damping \(b_{M1}\), \(b_{M2}\) and current constants \(k_{t1}\), \(k_{t2}\) from Eq. (6). The inputs (winding currents \(i_1\) and \(i_2\)), rotor angular positions \(\varphi _1\), \(\varphi _2\) and the horizontal position of beam TCP point \(x_n\) are recorded.

The equations of motion, taken into account for the identification analysis, are the first and the second of the set of Eq. (1). Introducing the forces in Eq. (2) and electromagnetic moments in Eq. (6) into those particular equations, it is obtained:

$$\begin{aligned} \begin{aligned} \left( \frac{k_L}{L_1}\xi _1+\frac{b_L}{L_1}\dot{\xi }_1\right) r_1&= k_{t1}i_1 - I_1\ddot{\varphi }_1 - b_{M1}\dot{\varphi }_1 \\ -\left( \frac{k_L}{L_2}\xi _2+\frac{b_L}{L_2}\dot{\xi }_2\right) r_2&= k_{t2}i_2 - I_2\ddot{\varphi }_2 - b_{M2}\dot{\varphi }_2 \\ \end{aligned} \end{aligned}$$
(19)

where unknown parameters \(k_{t1,2}\), \(I_{1,2}\) and \(b_{M1,2}\) are in linear combination. Other parameters and terms are known or could be derived from the measured data and its numeric time derivative. Numeric time derivative is amended with non-causal zero-phase low pass data filtering as in the previous case.

Data acquisition and further processing keeps the measurement sampling frequency \(f_s = 1kHz\). The Eq. (19) are expanded in the measurement time and following matrices are created:

$$\begin{aligned} \mathbb {U} = \begin{bmatrix} \left( k_L\frac{\xi _1(0)}{L_1(0)}+b_L\frac{\dot{\xi }_1(0)}{L_1(0)}\right) r_1\\ \vdots \\ \left( k_L\frac{\xi _1(n\Delta t)}{L_1(n\Delta t)}+b_L\frac{\dot{\xi }_1(n\Delta t)}{L_1(n\Delta t)}\right) r_1\\ -\left( k_L\frac{\xi _2(0)}{L_2(0)}+b_L\frac{\dot{\xi }_2(0)}{L_2(0)}\right) r_2\\ \vdots \\ -\left( k_L\frac{\xi _2(n\Delta t)}{L_2(n\Delta t)}+b_L\frac{\dot{\xi }_2(n\Delta t)}{L_2(n\Delta t)}\right) r_2\\ \end{bmatrix} \end{aligned}$$
(20)
$$\begin{aligned} \mathbb {V} = \begin{bmatrix} i_1(0)&{} - \ddot{\varphi }_1(0)&{} - \dot{\varphi }_1(0)&{} 0 &{} 0 &{} 0 \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ i_1(n\Delta t)&{} - \ddot{\varphi }_1(n\Delta t)&{} - \dot{\varphi }_1(n\Delta t)&{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} i_2(0)&{} - \ddot{\varphi }_2(0)&{} - \dot{\varphi }_2(0) \\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ 0 &{} 0 &{} 0 &{} i_2(n\Delta t)&{} - \ddot{\varphi }_2(n\Delta t)&{} - \dot{\varphi }_2(n\Delta t)\\ \end{bmatrix} \end{aligned}$$
(21)

with \(\Delta t = \frac{1}{f_s}\) and \(n+1\) measured points, as in the previous case. Unknown parameters are placed into the vector:

$$\begin{aligned} \begin{aligned} \mathbb {U}=\mathbb {V} \begin{bmatrix} k_{t1}\\ I_1\\ b_{M1}\\ k_{t2}\\ I_2\\ b_{M2}\\ \end{bmatrix}\\ \end{aligned} \end{aligned}$$
(22)

Vector of unknowns is solved using least squares method leading to pseudo inversion assembly.

$$\begin{aligned} \begin{aligned} \begin{bmatrix} k_{t1}\\ I_1\\ b_{M1}\\ k_{t2}\\ I_2\\ b_{M2}\\ \end{bmatrix} = \left( \mathbb {V}^T\mathbb {V}\right) ^{-1}\mathbb {V}^T\mathbb {U}\\ \end{aligned} \end{aligned}$$
(23)

Resulting moments of inertia \(I_1 = 7.98\cdot 10^{-4}kgm^2\), \(I_2 = 8.74\cdot 10^{-4}kgm^2\), current constants \(k_{t1} = 0.15\frac{Nm}{A}\), \(k_{t2} = 0.2\frac{Nm}{A}\), and dampings \(b_{M1} = 0.035Nms\), \(b_{M2} = 0.005Nms\) are verified in the simulation experiment, where the Eq. (19) has currents \(i_1\), \(i_2\) and beam TCP position \(x_n\) as inputs. Comparisons of simulated rotor angular positions with measurement are at Figs. 9 and 10. Behaviour of motor 1 is at Fig. 9, and behaviour of motor 2 is at Fig. 10. The graphs show the experimental (measured) data, then simulated data and their difference as absolute error. System drift at low frequencies (less than 0.3Hz) is subtracted.

Fig. 9
figure 9

Verification of identified model: motor parameters, motor 1 excitation

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Fig. 10
figure 10

Verification of identified model: motor parameters, motor 2 excitation

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The comparison in frequency domain is at Fig. 11. Left part shows behaviour of the first motor, right part shows the behaviour of the second motor.

Fig. 11
figure 11

Verification of identified model: motor parameters, frequency analysis

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2.2 System linearization

The set of Eq. (1) is linearized at the particular equilibrium operating point. The only nonlinear terms are in Eq. (4), the length \(L_1\) and \(L_2\) depend on variable \(x_n\), and the same is for angles \(\alpha _1\) and \(\alpha _2\). Horizontal displacement \(x_n\) is projected into the \(L_1\) and \(L_2\) directions as follows:

$$\begin{aligned} \begin{aligned} L_1 - L_{10}&= x_n \cos {\alpha _1}\\ L_2 - L_{20}&= - x_n \cos {\alpha _2}\\ \end{aligned} \end{aligned}$$
(24)

while change in \(\alpha _1\) and \(\alpha _2\) due to the small \(x_n\) is neglected. Cable longitudinal deformation in Eq. (3) is further simplified:

$$\begin{aligned} \begin{aligned} \xi _1&= x_n \cos {\alpha _1} + \varphi _1 r_1\\ \xi _2&= - x_n \cos {\alpha _2} - \varphi _2 r_2 \\ \end{aligned} \end{aligned}$$
(25)

The linearized system of eigen equations of motion is following:

$$\begin{aligned} \begin{aligned} I_1\ddot{\varphi }_1&= i_1k_{t1} - \frac{k_Lr_1}{L_1}\left( x_n \cos {\alpha _1} + \varphi _1 r_1\right) - \frac{b_Lr_1}{L_1}\left( \dot{x}_n \cos {\alpha _1} + \dot{\varphi }_1 r_1\right) - b_{M1}\dot{\varphi }_1 \\ I_2\ddot{\varphi }_2&= i_2k_{t2} - \frac{k_Lr_2}{L_2}\left( x_n \cos {\alpha _2} + \varphi _2 r_2 \right) - \frac{b_Lr_2}{L_2}\left( \dot{x}_n \cos {\alpha _2} + \dot{\varphi }_2 r_2 \right) - b_{M2}\dot{\varphi }_2 \\ m\ddot{x}_n&= -k(x_n-x) - b(\dot{x}_n-\dot{x}) +\frac{k_L}{L_2}\left( - x_n \cos {\alpha _2} - \varphi _2 r_2 \right) \cos {\alpha _2}+\\ {}&+\frac{b_L}{L_2}\left( - \dot{x}_n \cos {\alpha _2} - \dot{\varphi }_2 r_2 \right) \cos {\alpha _2} -\frac{k_L}{L_1}\left( x_n \cos {\alpha _1} + \varphi _1 r_1\right) \cos {\alpha _1}-\\ {}&-\frac{b_L}{L_1}\left( \dot{x}_n \cos {\alpha _1} + \dot{\varphi }_1 r_1\right) \cos {\alpha _1}\\ \end{aligned} \end{aligned}$$
(26)

with inputs \(i_1\), \(i_2\) and cart coordinate x (with its time derivative \(\dot{x}\)).

3 Vibration damping analysis

The sensitivity of real mechanical systems to external disturbance determines their usefulness and suitability for particular task. Analysis and reduction of sensitivity are the main goal of this paper.

External disturbance with significant effect of given presented mechanical system (Fig. 1) is the horizontal force application on the beam tip (TCP). For evaluation of system vibration damping capability, the frequency analysis of the transfer function \(\frac{Y}{W}\) is performed. W represents disturbance and Y system output, in the case of our system the disturbance is applied horizontal force and output is the horizontal coordinate of the beam tip, \(x_n\).

The state-space representation is used for system properties evaluation. The comparison is performed also with the system without the cables. The state-space system description has the following form:

$$\begin{aligned} \begin{aligned} \dot{\mathbf {z}}&= \mathbb {A}\mathbf {z} + \mathbb {B}\mathbf {u}\\ \mathbf {y}&= \mathbb {C}\mathbf {z} + \mathbb {D}\mathbf {u}\\ \end{aligned} \end{aligned}$$
(27)

The state-space matrices for the system without cables excited by external force w(t) as at Fig. 12 are the following:

$$\begin{aligned} \begin{aligned} \mathbb {A}&= \begin{bmatrix} 0&{} -\frac{k}{m}\\ 1 &{} -\frac{b}{m}\\ \end{bmatrix}\ \ \ \ \ \ \ \ \mathbb {B} = \begin{bmatrix} \mathbb {B}_x &{} \mathbb {B}_w\\ \end{bmatrix}\ \ \ \ \mathbb {B}_x = \begin{bmatrix} \frac{k}{m}\\ \frac{b}{m}\\ \end{bmatrix}\ \ \ \ \mathbb {B}_w = \begin{bmatrix} \frac{1}{m}\\ 0\\ \end{bmatrix}\\ \mathbb {C}&= \begin{bmatrix} 0 &{} 1\\ \end{bmatrix}\ \ \ \ \mathbb {D}= \begin{bmatrix} 0 &{} 0\\ \end{bmatrix} \end{aligned} \end{aligned}$$
(28)

The output is \(x_n\) coordinate, inputs are x and w:

$$\begin{aligned} \begin{aligned} \mathbf {z} = \begin{bmatrix} z_1\\ z_2\\ \end{bmatrix}\ \ \ \ z_2 = x_n\ \ \ \ \mathbf {u} = \begin{bmatrix} x\\ w\\ \end{bmatrix}\ \ \ \ \mathbf {y} = \begin{bmatrix} x_n \end{bmatrix} \end{aligned} \end{aligned}$$
(29)

The system is transformed into the Laplace domain and corresponding transfer function \(\frac{X_n}{W}\) for the system without cables is

$$\begin{aligned} \begin{aligned} \frac{X_n}{W}&= \mathbb {C}\left( \mathbb {I}s-\mathbb {A}\right) ^{-1}\mathbb {B}_w=\\&= \frac{1}{m s^2+b s + k} \end{aligned} \end{aligned}$$
(30)

where symbol s represents Laplace variable, \(X_n\) and W are output (beam tip horizontal coordinate) and input (horizontal force) in the Laplace domain, and \(\mathbb {I}\) represents eigenmatrix of the same size as matrix \(\mathbb {A}\).

3.1 Effect of cables with drives in position control mode

The system with the cables and drives in position control mode with external disturbance force w(t) is depicted at Fig. 12. If the drives are in position control mode, the prescribed drive positions form in fact the kinematic excitation. The system properties are compared with the system without cable structure.

Fig. 12
figure 12

Model with drives in position control mode (kinematic constraint is applied at angles \(\varphi _1\) and \(\varphi _2\)) and applied external disturbance force w

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The linearized equations describing the system in Fig. 12 are derived from Eq. (26), and only the third equation is used. For obtaining the frequency response, the state space representation is assembled.

$$\begin{aligned} \begin{aligned} \mathbb {A}_1&= \begin{bmatrix} 0&{} -\frac{1}{m}\left( k+\frac{k_L\cos ^2{\alpha _1}}{L_1}+\frac{k_L\cos ^2{\alpha _2}}{L_2}\right) \\ 1 &{} -\frac{1}{m}\left( b+\frac{b_L\cos ^2{\alpha _1}}{L_1}+\frac{b_L\cos ^2{\alpha _2}}{L_2}\right) \\ \end{bmatrix}\\ \mathbb {B}_1&= \begin{bmatrix} \mathbb {B}_\varphi &{}\mathbb {B}_{x1} &{} \mathbb {B}_{w1}\\ \end{bmatrix}\\ \mathbb {B}_\varphi&= \begin{bmatrix} -\frac{k_Lr_1\cos {\alpha _1}}{mL_1} &{} -\frac{k_Lr_2\cos {\alpha _2}}{mL_2} \\ -\frac{b_Lr_1\cos {\alpha _1}}{mL_1}&{} -\frac{b_Lr_2\cos {\alpha _2}}{mL_2} \\ \end{bmatrix}\ \ \ \ \mathbb {B}_{x1} = \begin{bmatrix} \frac{k}{m}\\ \frac{b}{m}\\ \end{bmatrix}\ \ \ \ \mathbb {B}_{w1} = \begin{bmatrix} \frac{1}{m}\\ 0\\ \end{bmatrix}\\ \mathbb {C}_1&= \begin{bmatrix} 0 &{} 1\\ \end{bmatrix}\ \ \ \ \mathbb {D}_1 = \begin{bmatrix} 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix} \end{aligned} \end{aligned}$$
(31)

The output is \(x_n\) coordinate, and inputs are \(\varphi _1\), \(\varphi _2\), x, and w.

$$\begin{aligned} \begin{aligned} \mathbf {z}_1 = \begin{bmatrix} z_{11}\\ z_{12}\\ \end{bmatrix}\ \ \ \ x_n = z_{12}\ \ \ \ \mathbf {u}_1 = \begin{bmatrix} \varphi _1\\ \varphi _2\\ x\\ w\\ \end{bmatrix}\ \ \ \ \mathbf {y}_1 = \begin{bmatrix} x_n \end{bmatrix} \end{aligned} \end{aligned}$$
(32)

The transfer function \(\frac{X_n}{W}\) is

$$\begin{aligned} \begin{aligned} \frac{X_n}{W}&= \mathbb {C}_1\left( \mathbb {I}s-\mathbb {A}_1\right) ^{-1}\mathbb {B}_{w1}=\\&= \frac{1}{m s^2+\left( b + \frac{b_L\cos ^2{\alpha _1}}{L_1} + \frac{b_L\cos ^2{\alpha _2}}{L_2}\right) s + k +\frac{k_L\cos ^2{\alpha _1}}{L_1}+ \frac{k_L\cos ^2{\alpha _2}}{L_2} } \end{aligned} \end{aligned}$$
(33)

The original system (without cables) has eigenfrequency 10.81Hz and damping \(\zeta =4.6\cdot 10^{-3}\), system with cables has eigenfrequency 21.27Hz with \(\zeta =0.4\), and we are going to introduce the state feedback control with damped arbitrarily prescribed poles (\(\Omega =30Hz\) and \(\zeta =0.8\)) using pole placement algorithm [46]. The system \(\left( \mathbb {A}_1,\mathbb {B}_\varphi \right)\) is controllable and feedback gain \(\mathbb {K}_{place1}\) is obtained. The transfer function of controlled system Eq. (34) is investigated too.

$$\begin{aligned} \begin{aligned} \frac{X_n}{W}&= \mathbb {C}_1\left( \mathbb {I}s-\left( \mathbb {A}_1-\mathbb {B}_\varphi \mathbb {K}_{place1}\right) \right) ^{-1}\mathbb {B}_{w1}\\ \end{aligned} \end{aligned}$$
(34)

The comparison of external disturbance sensitivity is shown at Fig. 13. System extended with cables move the overall eigenfrequency and its behaviour is in comparison with original structure more damped. If we use the state feedback to choose the eigenfrequency and adequate damping, the sensitivity to disturbance further decreases.

Fig. 13
figure 13

Magnitude phase diagram of the transfer function \(\frac{x_n}{w}\) is plotted, \(x_n\) is beam tip horizontal coordinate, and w is horizontal disturbance force applied at beam tip. There is performed the comparison between original beam structure (blue line) and structure with cables with drives in position control mode (without control — red line and with state feedback control — yellow line)

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The simulation experiment is performed too and it verifies the findings. The system with and without cables is loaded with disturbance force at different frequencies (sweep cosine function is used) to obtain the \(\frac{x_n}{w}\) transfer function behaviour. At Fig. 13, the dotted line represents the simulation experiment verification of system with pole placement.

3.2 Effect of cables with drives in torque control mode

The full model of experimental stand on Fig. 2 excited at beam tip with external horizontal force w(t) as disturbance is investigated. The system with cables is compared with the system without cable structure, which is common with the previous case.

The linearized equations at Eq. (26) are used for assorting the state space representation. It further serves as a base for the system frequency response.

$$\begin{aligned} \begin{aligned} \mathbb {A}_2&= \begin{bmatrix} 0&{} -\frac{1}{m}\left( k+\frac{k_L\cos ^2{\alpha _1}}{L_1}+\frac{k_L\cos ^2{\alpha _2}}{L_2}\right) &{} 0 &{}-\frac{k_Lr_1\cos {\alpha _1}}{I_1L_1} &{} 0 &{}-\frac{k_Lr_2\cos {\alpha _2}}{I_2L_2}\\ 1 &{} -\frac{1}{m}\left( b+\frac{b_L\cos ^2{\alpha _1}}{L_1}+\frac{b_L\cos ^2{\alpha _2}}{L_2}\right) &{} 0 &{}-\frac{b_Lr_1\cos {\alpha _1}}{I_1L_1} &{} 0 &{}-\frac{b_Lr_2\cos {\alpha _2}}{I_2L_2}\\ 0 &{} -\frac{k_Lr_1\cos {\alpha _1}}{mL_1}&{} 0 &{} -\frac{k_Lr_1^2}{I_1L_1} &{} 0 &{} 0\\ 0 &{} -\frac{b_Lr_1\cos {\alpha _1}}{mL_1}&{} 1 &{} -\frac{b_{M1}}{I_1}-\frac{b_Lr_1^2}{I_1L_1} &{} 0 &{} 0\\ 0 &{} -\frac{k_Lr_2\cos {\alpha _2}}{mL_2} &{} 0 &{} 0 &{} 0 &{} -\frac{k_Lr_2^2}{I_2L_2}\\ 0 &{} -\frac{b_Lr_2\cos {\alpha _2}}{mL_2} &{} 0 &{} 0 &{} 1 &{} -\frac{b_{M2}}{I_2}-\frac{b_Lr_2^2}{I_2L_2}\\ \end{bmatrix}\\ \mathbb {B}_2&= \begin{bmatrix} \mathbb {B}_i &{}\mathbb {B}_{x2} &{} \mathbb {B}_{w2}\\ \end{bmatrix}\\ \mathbb {B}_i&= \begin{bmatrix} 0 &{} 0\\ 0 &{} 0\\ k_{t1} &{} 0\\ 0 &{} 0\\ 0 &{} k_{t2}\\ 0 &{} 0\\ \end{bmatrix}\ \ \ \ \mathbb {B}_{x2} = \begin{bmatrix} k\\ b\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}\ \ \ \ \mathbb {B}_{w2} = \begin{bmatrix} 1\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}\\ \mathbb {C}_2&= \begin{bmatrix} 0 &{} \frac{1}{m} &{} 0 &{} 0 &{} 0 &{} 0\\ \end{bmatrix}\ \ \ \ \mathbb {D}_2 = \begin{bmatrix} 0&0&0&0 \end{bmatrix} \end{aligned} \end{aligned}$$
(35)

The output is \(x_n\) coordinate, inputs are electric drive currents \(i_1\) and \(i_2\), cart coordinate x, and external force w.

$$\begin{aligned} \begin{aligned} \mathbf {z}_2 = \begin{bmatrix} z_{21}\\ z_{22}\\ z_{23}\\ z_{24}\\ z_{25}\\ z_{26}\\ \end{bmatrix}\ \ x_n = \frac{z_{22}}{m}\ \ \varphi _1 = \frac{z_{24}}{I_1}\ \ \varphi _2 = \frac{z_{26}}{I_2}\ \ \mathbf {u}_2 = \begin{bmatrix} i_1\\ i_2\\ x\\ w\\ \end{bmatrix}\ \ \mathbf {y}_2 = \begin{bmatrix} x_n \end{bmatrix} \end{aligned} \end{aligned}$$
(36)

The transfer function \(\frac{X_n}{W}\) as a measure of vibration damping is

$$\begin{aligned} \begin{aligned} \frac{X_n}{W}&= \mathbb {C}_2\left( \mathbb {I}s-\mathbb {A}_2\right) ^{-1}\mathbb {B}_{w2}=\\&= \frac{c_4s^4+c_3s^3+c_2s^2+c_1s+c_0}{s^6+a_5s^5+a_4s^4+a_3s^3+a_2s^2+a_1s+a_0} \end{aligned} \end{aligned}$$
(37)

where coefficients \(a_0,...,a_5\) and \(c_0,...,c_4\) are in the Appendix section.

The model with drives and cables has three eigenfrequencies: 6.87Hz with damping 0.2, 15.78Hz with damping 0.5, and 24.95Hz with damping 0.6. The first eigenfrequency, which appears due to the electric drives equations of motion, is danger one, because of the relatively small damping. The behaviour of the system at Fig. 14 affirms the suspect. That is the reason to introduce state feedback control as in the previous case. The system \(\left( \mathbb {A}_2,\mathbb {B}_i\right)\) is controllable and feedback gain \(\mathbb {K}_{place2}\) is calculated. The prescribed poles with arbitrarily prescribed frequencies are 30Hz, 80Hz, and 90Hz with common damping 0.8. The transfer function of controlled system is the following:

$$\begin{aligned} \begin{aligned} \frac{X_n}{W}&= \mathbb {C}_2\left( \mathbb {I}s-\left( \mathbb {A}_2-\mathbb {B}_i\mathbb {K}_{place2}\right) \right) ^{-1}\mathbb {B}_{w2}\\ \end{aligned} \end{aligned}$$
(38)

The transfer function \(\frac{x_n}{w}\) behaviour as a measure of system sensitivity to external disturbances is shown at Fig. 14. System extended with cables has similar behaviour as the original system at low frequencies, but newly arises the first eigenfrequency and the feedback in this region is needed to improve the system vibration suppression properties.

Fig. 14
figure 14

Magnitude phase diagram of the transfer function \(\frac{x_n}{w}\) is plotted, \(x_n\) is beam tip horizontal coordinate, and w is horizontal disturbance force applied at beam tip. There is performed the comparison between original beam structure (blue line) and structure with cables with electric drives in torque control mode (without control — red line and with state feedback control — yellow line)

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Similarly as in the first case, the simulation experiment is performed and it verifies the findings. At Fig. 14, the dotted line represents some of the simulation experiment points.

4 Experimental test

The experimental verification of proposed cable system is performed. The setup is described in Fig. 15.

Fig. 15
figure 15

Experimental stand description

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The mechanical system consists of the cart mounted to linear axis Rexroth with the electric drive Kollmorgen 6SM 57M-3.000-S3-1325. On the cart is mounted a beam with the cables attached at its tip. The cables are actuated with electric drives Kollmorgen D062M-92-9310-010. The cart and beam tip horizontal positions are measured by lasertracker Leica AT960-MR and Leica AT901-B, respectively, and the drive angular positions are measured by internal sensors. The whole system is managed by real-time control system Beckhoff AX5911 TwinCAT3 based on bus EtherCAT. The horizontal external force at the beam tip is excited by shaker Tira TV51140 and measured by force sensor PCB Piezotronics 208C02 connected to the Brüel &Kjaer 2694 amplifier.

The external horizontal force, as a disturbance, is set to be sweep cosine function with frequency change from 0.5 until 20Hz. The force magnitude is tuned manually.

Measurement result with feedback control algorithm shows the transfer function \(\frac{x_n}{w}\) at Fig. 16. The feedback gain is obtained in a similar way as for Eq. (38) with prescribed frequencies 30Hz, 35Hz, and 40Hz and common damping 0.5; the condition of always loaded cables is fulfilled by demanded torque modification. The cable structure with applied feedback control decreases the system sensitivity to the external disturbance up to \(-20dB\) and the system eigenfrequency is not significantly excited.

Fig. 16
figure 16

Magnitude phase diagram of the transfer function \(\frac{x_n}{w}\) is plotted, \(x_n\) is measured beam tip horizontal position, and w is horizontal disturbance force measured at beam tip. The comparison between original beam structure (blue line) and structure with cables with preload torque and feedback control (red line)

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5 Conclusions

Vibration suppression is frequently solved in engineering practice. The presented approach deals with the flexible structure and such systems are often problematic. The flexibility does not allow higher dynamic movements without residual vibrations. To overcome this problem, the auxiliary cable structure is attached to the system to increase system stiffness and remove the oscillations.

The resonance amplitude of the plant was suppressed and behaviour of the system until 15Hz was reduced by \(-20dB\) using the cable structure with feedback control. It was shown that the concept is suitable to be used for vibration damping of slender structures.

Obtained results conceptionaly prove the validity of proposed concept. Follow-up research will deal with deeper experimental verification of the achieved results.

The obtained results can be summarized as follows:

  • Experimental stand representing the main properties of machine tools with a large workspace and long and heavy movable parts damped by auxiliary cable structure is prepared

  • The cables can be easily reconfigured and the original workspace of machine is almost preserved

  • Experimental stand identification is performed and model is created

  • State feedback simplified model control law for cable structure is assembled

  • Frequency analysis shows the vibration suppression

  • Low frequencies are suppressed, especially first eigenfrequency

  • Additional cable system uses standard drives commonly used by machine tool manufacturers

  • Experimental test shows the desired behaviour with transfer function reduction of 20dB