# Research of lifting equipment dynamics

## Vytautas Kazimieras Augustaitis1, Vladimir Gican2, Arunas Jakstas3, Bronislovas Spruogis4, Vytautas Turla5

1, 2, 3, 5Vilnius Gediminas Technical University, 28 Basanaviciaus str., LT-03224, Vilnius, Lithuania

4Vilnius Gediminas Technical University, Plytines 27, LT-10105, Vilnius, Lithuania

5Corresponding author

Journal of Vibroengineering, Vol. 16, Issue 4, 2014, p. 2082-2088.
Received 26 February 2014; received in revised form 6 June 2014; accepted 12 June 2014; published 30 June 2014

Copyright © 2014 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

The mechanical properties of the rope – stiffness and internal friction changes during the movement of cargo. For this reason, under the influence of gravity and inertial forces in the rope having varying tension. Thus, the lift mechanism further exposed to dynamic loads. The highest value of this load is obtained at the beginning of lifting and lowering in place. The mathematical model of the lift mechanism was investigated using the original software package and found dynamic load. In modern control system used cranes lifting mechanism, taking into account the deviation from the vertical rope. Dynamic load on the rope can distort the signals. Results are used to improve the control system of the lifting mechanism.

Keywords: lifting equipment, rope’s elasticity, dynamic load factor.

#### 2. Mathematical model and the program for its computing

In Fig. 1, a dynamic model of a cargo lifting mechanism using overhead crane is provided. Rope with a hook, where cargo of mass $M$ is suspended, is fixed to a rope pulley in point $O$. Rope pulley can move under impact of gear on crane rails on ${X}_{0}$ coordinate. Crane with a rope pulley can move under impact of gear on ${Y}_{0}$ coordinate using support rails. Gears are used to set, by using e.g. controllers, given speed on the axes. The same goes for the length of rope. Diagrams on speeds set during modelling are provided in Fig. 3. We shall assume that the performance mechanisms use high precision in working out given movement, which allows for application of cinematic agitation on ${X}_{0}$, ${Y}_{0}$ and ${L}_{0}$ coordinates. Length of rope L can change under control, force of gravity and dynamic loads.

Fig. 1. The dynamic model The following expressions for kinetic and potential energy and dissipative function were established. Having differentiated Lagrange equations, equations were received for generic ${\mathrm{\Theta }}_{x}$, ${\mathrm{\Theta }}_{y}$, and ${\delta }_{L}$ coordinates. General solution can be presented as follows:

(1)
(2)
(3)

here: $L$ – the length of the rope: $L={L}_{0}+{\delta }_{L}$, ${L}_{0}$– the length of the rope without a cargo predetermined by a position of the rope drum, ${\delta }_{L}$– the elongation of the rope caused by a cargo, ${C}_{1}={C}_{0}+k{\delta }_{L}$, ${C}_{0}$ – initial relative stiffness of the rope (stiffness of 1 m of the rope), $k$ – ratio estimating stiffness increase by rope strain.

Fig. 2 demonstrates Simulink-diagram of lifting mechanism model based on 1-3 expressions.

Fig. 2. Model (Simulink-diagram) of lifting mechanism #### 3. Terms of simulation

Values of parameters in the model appear in Table 1, while the values of agitation signals appear in Fig. 3. Damping data corresponds to .

Table 1. The parameters of the model

 $G$ 9.81 m/s2 Free fall (gravitational) acceleration ${L}_{0}$ 10 m Initial length of the rope $M$ 950 kg Mass of cargo ${H}_{1}$ 225 N·s/m Damping in the rope ${H}_{2}$ 1050 Nm·s/rad Damping in the cargo suspension point ${C}_{0}$ 0,28×107 N Initial relative tensile stiffness of the rope $k$ 0,122×108 N/m Ratio estimating stiffness increase by rope strain

Fig. 3. Diagrams of change in pre-set speeds of lifting mechanism (DL0), rope pulley (DX0) and crane (DY0) #### 4. The results of the investigation and discussion

Processing of modelling results employs the following correlations between key and auxiliary coordinates:

1. Projections of distances from suspension point on rope pulley to cargo centre of gravity:

(4)
${X}_{m}={\Theta }_{x}L,$
(5)
${Y}_{m}={\Theta }_{y}L.$

2. Distance from starting point of movement to cargo centre of gravity and height of hook over ground level:

(6)
${X}_{M}={X}_{0}+{X}_{m},$
(7)
${Y}_{M}={Y}_{0}+{Y}_{m},$
(8)
${h}_{kr}={h}_{k}-L,$

where ${h}_{k}$ – starting height of hook position, ${h}_{kr}$ – current height of hook.

Fig. 4 provides trajectories of key coordinates in the model based on modelling session.

Fig. 4. Changes in angles of swinging ${\theta }_{x}$, ${\theta }_{y}$ in cycle of movement of cargo As demonstrated in Fig. 4 in case of horizontal movement of cargo suspended on hook, swing occurs.

Fig. 5 demonstrates change in location of centre of cargo gravity in movement cycle.

Fig. 5. Trajectory of cargo movement given operation of lifting mechanism and taking into account swinging: a) on XM coordinate; b) on YM coordinate; c) view of trajectory from above; d) view of trajectory in space: 1 – actual trajectory taking into account cargo swinging, 2 – trajectory set by movement of crane mechanisms a) b) c) d)

Fig. 5 demonstrates that swinging can materially distort trajectory of movement of cargo, which in turn can have severe consequences. Amplitude of swinging changes in course of movement of freight and depends, e.g. on scope and nature of horizontal acceleration at cargo suspension point, length of rope and other factors.

Fig. 6 displays nature of cargo swinging at the end of movement cycle, when engines of horizontal movement are switched off, and the winch of rope pulley brings down the cargo.

Fig. 6. Nature of swinging of cargo, when winch of rope pulley operates on lowering only: a) components ${X}_{m}$ (solid line) and ${Y}_{m}$ (dotted line), b) view of trajectory in space a) b)

Fig. 7. Oscillations of stretching effort on rope: a) general picture; b) by selection of ease of rope; c) by acceleration of gear of $X$ axis; d) by breaking of gears in $X$ and $Y$ axes; e) by acceleration of lifting mechanism; f) linear stretch of rope ${\delta }_{L}$ a) b) c) d) e) f)

Notably the amplitude of swinging in case of lowering cargo (Fig. 6) increases. In the meantime however one must stabilise position of cargo before placing the same on the foundation. In order to stabilise the position one must weight for a certain time or take special means to tackle swinging. Amplitude of swinging can be reduced by making use of special laws  of smooth acceleration and breaking of horizontal movement as well as by applying other laws on optimum movement, e.g. by using soft logic  and return correlation on deviation of cargo rope from vertical position.

Fig. 7 demonstrates results of modelling of longitudinal load of rope in the cycle of movement of cargo. Different events, e.g. sudden selection of ease, turning on and off controllers of horizontal movement, swinging of cargo; all cause change in longitudinal force, which affects the rope and frequencies of oscillations [19, 20].

Fig. 8 demonstrates changes in frequency of proper oscillations of cargo suspended on an elastic rope as operational length of rope changes. This can be explained by the fact that longitudinal stiffness of rope (for stretching) decreases as the length increases. The model has demonstrated increase in frequency at 1.38 times while the length of rope was reduced by 1.9 times.

Fig. 8. Dependence of frequency of cargo oscillations when suspended on a rope from operational length of the rope Fig. 9. Rope stiffness dependence $k$ from rope elongation ${\delta }_{L}$ Preparing data for modelling the investigation of rope strain was performed. The results of the investigation are presented in Fig. 9. The increase of rope strain is estimated by rate $k$.

As a result of modelling session trajectories of key coordinates were obtained in cycle of movement of cargo. Analysis of dependences obtained allows for stating that inertia forces in the start and end of transportation make delivery of cargo more difficult given strict requirements on precision positioning. This must be taken into account when designing mechatronic control software.

#### 5. Conclusions

1) In the beginning of the lifting process, additional dynamic loads appear. The value of the dynamic load factor achieves up to 1.7. The duration of the impact – up to 10 s.

2) On cargo swinging, dynamic loads appear in the rope. The value of the dynamic load factor depends on the amplitude of swinging. On appearance of swinging, the frequency of the dynamic loads is not bound with the frequency of swinging, but depends on the stiffness of the rope and mass of the cargo. Later, the frequency of the dynamic loads becomes equal to the frequency of swinging and they act within the whole period of swinging.

3) The dynamic load of the rope in begging of swinging is significantly bigger in comparison with ones during further transportation.

4) On cargo descending, the amplitude of cargo swinging increases: the precision of cargo delivery decrease as well.

5) The amplitude of the swing depends on the nature and horizontal accelerations of the suspension point, load rope length and other factors. The amplitude of the swing can be reduced by smooth acceleration and deceleration of horizontal displacement. The swing can be significantly reduced by reversing the gear of displacement at the end of transportation. The most effect can be get using combination equal acceleration and deceleration.

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