1 Introduction

The dynamic behavior of the crane beams depends on the moving loads on them. This problem, which is called a moving load problem, has a wide place in engineering compliance. With the developing technology, changes in design conditions and parameters are inevitable. Today's cranes are generally designed by neglecting their dynamic effects due to their low-speed operation. However, with the increasing technology, production speeds have increased and the need for fast-moving cranes is increasing to move the products from one place to another, especially in the loading and unloading of cargoes in ports. The dynamic behavior of the crane beam system must be precisely determined at the design stage to determine the service life of the crane that will operate quickly under heavy conditions. The dynamic behavior of the structures subjected to moving load is considered an important problem by the researchers and the studies carried out on this subject.

Fryba carried out theoretically the most comprehensive study on the vibration of structures and solids by the effect of loads acting on the beam [1]. Low performed vibration analysis using self-functions for beams carrying multiple masses. Mode analysis with a polynomial approximation is given in comparison with both analytical and experimental results. The comparison results gave valid results for the proposed models [2]. Oguamanamand has modeled a crane consisting of a simple supported uniform Euler–Bernoulli beam and a load. The crane carrier and load connection are modeled as suspended with a massless bar, boundary conditions are determined to be subjected to gravitational force and move along the beam direction. Using the Hamilton principle, two pairs of integrodifferential equations for crane motion are derived. For the beam-crane system, natural frequencies of vibrations were determined, and analytical frequency equations were derived for each case. Numerical studies were performed according to car walking speed, mass of useful load, mass of car, suspended useful load length. It was determined that the maximum collapse on the beam due to the inertia of the car occurred at the end of the beam at high speeds and in the middle of the beam at slow speeds [3, 4]. Wu has examined the dynamic behavior of structures under moving loads by analytical methods and finite element methods. In previous studies, moving masses were considered as motion loads without considering the effects of inertia. This is usually not the right method, where inertial effects affect the analysis results. First, the movement of the single mass on the simply supported beam is modeled and a methodology is developed. The developed methodology was applied to the traveling crane portal crane system and experimentally the dynamic behavior of the system was examined [5]. Wu et al. have used standard packet analysis programs to determine the dynamic behavior of structures under time-dependent loads. Firstly, the analyzes were sampled by performing the single mass moving on a single beam, and then the same methodology was repeated for the mobile portal crane moving in two dimensions [6].

Yang et al. have been considered a tower crane model, the system has been modeled by taking into account the spherically shown beneficial load carried by pendulum movement on the freely articulated beam with a rotatable and movable carrier. The mathematical model was created according to the Euler–Bernoulli beam theory. The useful load is modeled as a point mass and connected to a free supported beam that can be rotated by a massless elastic cable. According to the Hamilton principle, motion loads are taken into and out of the beam plane and motion equations are obtained [7]. Abu Hilal [8] has done studies on the forced vibration dynamic of the Euler–Bernoulli beam based on the Green functions. Wu [9] carried out the dynamic analysis of the 3D portal crane by using the finite element method and experimental. In another study on the control of oscillations of non-linear crane loads, a mechanical filter concept has been developed to control oscillations [10]. Ju [11] has studied the dynamic response of the tower crane triggered by the pendulum movement of the load. Abdel Rahman et al. [12] examined crane models and crane control strategies in the literature and proposed appropriate model and control criteria for crane. Zirnic et al. [13] have tried to explain the effects of an elastically coupled moving object on the gantry crane by analytical and finite element method. Gasic et al. [14] have studied a mass approach to approximate the dynamic response of a flexible L-shaped structure with the lumped mass subjected to a moving load. Ouyang summarized the studies on this subject by conducting a survey on the dynamic effects of structures under moving loads. Various basic concepts specific to the Moving load problem acting on a simply supported beam are introduced and necessary mathematical expressions are presented [15]. Şimşek [16] has studied the vibration of a functionally graded simple supported beam according to Euler Bernoulli and Timoshenko beam theories. Pesterev et al. [17] have studied the dynamic response of elastic beams that make many movable oscillations and have shown that oscillations on system vibrations make the system's vibration dynamics even more complex. Pesterev and Bergman [18] developed a new solution method for the system, which is also a moving oscillator on the beam, and showed its effectiveness with numerical solutions.

Gasic [19] also studied the dynamic response of the beam according to different spring coefficients and different velocities bearing the oscillator and examined their effects on the system. Kıral et al. [20] Used air as a moving load on a cantilever beam and conducted an experimental study by measuring the dynamic displacement response at different points of the beam with laser displacement sensors. Esen [21] the dynamic response of a mass moving on the beam with the finite element approach has made studies on obtaining the dynamic response and demonstrated the effectiveness of the method. Yıldırım and Esim [22] have examined the dynamic effects that occur at the midpoints of the beam according to different working speeds in the case of a beam or double beam in the transport of the same load, using the finite element method. Kıral et al. have investigated the experimental analysis of free and forced vibrations of a cantilever symmetric laminated composite beam with different layup sequences. The results obtained show that the lay-up sequence has an important role in the dynamic response of the laminated composites and the velocity of the moving load affects the dynamic response [23].

Yıldırım and Esim have investigated free vibration and harmonic analysis of multi carriage double bridge overhead crane systems using finite element analysis [24, 25]. Also, they have worked on vibration analysis of an experimental double bridge crane system with artificial neural networks [26]. Modal testing and harmonic analysis are a technique which has been widely used in structural engineering for determining structural characteristics, such as natural frequencies, mode shapes, Frequency response function, etc. Many researchers have studied the relating problems. For example, Gasic and Petkovic [27] have studied the determination of eigenfrequencies of overhead cranes in the vertical plane. Yıldırım and Esim [28] modeled a drill column machine and performed harmonic vibration analysis through the finite element method. Yongfeng Zhang et al. [29] have studied the dynamics of the four-link combination boom portal crane. Pinca et al. [30] have analyzed the stresses and deformations of the overhead type crane. Esim and Yıldırım [31] have investigated the dynamic characteristics of the drill machine structure.

Unlike the above-mentioned studies, the contributions of the paper can be summarized as:

  1. 1.

    In crane systems, dynamic effects occur as a result of the continuous position change during lifting and transporting the load. Especially when considering double girder bridges in crane systems; The structure is designed symmetrically, and dynamic analyzes are carried out by considering the effects on the beams as single beams

  2. 2.

    The crane system, lifting group, rope, and load move together. There is a difference in the center of gravity of the load because the lifting group is not structurally symmetrical, and most of the transported loads are not symmetrical. Different dynamic effects occur on the crane due to oscillations due to the connection of the load with the rope to the cart.

  3. 3.

    Also, as can be seen from the modal analysis, different dynamic effects occur in the beams due to the flexibility of the beams in the crane system [24, 25]. At the same time, studies on vertical vibrations in suspension-type bridges have been reviewed and flexibility has been shown to have a significant effect on beams [32]. One of the important contributions of this study is the flexible modeling of the beams

  4. 4.

    The calculation of the contact forces on the beams in this case. Therefore, it is a more eastern approach to perform analysis for cases where all these effects are taken into consideration.

In this study, to calculate all these effects during the joint movement of the load and trolley in the crane system, the ADAMS program modeled the beams as flexible and the rope connecting the car and load as spring and simulated the actual working conditions. Thus, all inertial effects are included in the solution. The contact forces acting on different car numbers were obtained by the ADAMS program and these results were analyzed in the ANSYS program according to time.

Finally, the results were compared with the experimental results. Compared with integrated solutions and field measurement results, it is seen that the computational accuracy of the proposed method has achieved considerable experimental results.

The organization of the paper can be written as follows; Sect. 2 gives the theory of dynamic behavior of bridge cranes subjected to moving load. In Sect. 3, the new proposed numerical approach of dynamic analysis of bridge cranes is outlined in detail. Section 4 presents an experimental study. Simulation and experimental results are given in Sect. 5. Finally, the paper is concluded with Sect. 6.

2 Dynamic analysis of bridge crane subjected to moving load

In this study, the dynamic effects that occur when the carriage moves on the bridge were investigated. The schematic view on which the car moves on the crane system is given in the figure below. The general approach to the dynamic analysis of crane systems is to take into account the fact that the crane is divided into two parts as fixed and moving parts. The schematic view where the load and the carriage move is given in Fig. 1.

Fig. 1
figure 1

Schematic representation of moving loads the dynamic model

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At this stage, firstly, the solution is made for a single-car situation. Here, the beam consists of the Euler Bernoulli beam and the load carried by the moving carriages and connected to the carriages with the rope [33].

2.1 Theoretical model of the bridge for a trolley system

In the model given in Fig. 2, considering the system under the effect of the moving load on which a car moves on the crane, the homogeneous simple supported Euler–Bernoulli beam is considered as a fixed element. While the car moving and the load moving on it can be considered as a constant force moving with velocity v in the vertical direction. The beam is assumed to be uniform and simply supported at both ends. The mechanical parameters for this beam are defined as: Elasticity modulus E, density ρ of the beam, beam cross-sectional area (box profile) A, Span L, and cross-section moment of inertia I.

Fig. 2
figure 2

a A constant force moving with velocity v in the vertical direction on bridge beam b Equivalent nodal forces of the element s subjected moving load

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The equation of motion of a multi-degree of freedom system divided into elements according to the finite element method is expressed as follows: [34],

$$\left[ m \right]_{{}} \left\{ {\ddot{u}} \right\} + \left[ c \right]_{{}} \left\{ {\dot{u}} \right\} + \left[ k \right]_{{}} \left\{ u \right\} = \left\{ {F(t)} \right\}$$
(1)

In this structure; \(\left[ M \right]\) is expressed as the mass matrix, \(\left[ C \right]\) is the damping matrix, \(\left[ K \right]\) is stiffness matrix. \(\left\{ {\ddot{u}} \right\}\) the acceleration vector of the whole structure, \(\left\{ {\dot{u}} \right\}\) the velocity vector of the whole structure, \(\left\{ u \right\}\) the displacement vector of the whole structure. \(\left\{ {F(t)} \right\}\) is defined as a time-dependent external force vector performed externally to the system.

When a vertical force (F) is applied on the beam, the external forces in all nodes except the nodes of the elements (Fig. 2b) that are exposed to the total force are zero. According to Clough and Penzien [38], the specified external force vector is defined in Eq. 2.

$$\left\{ {F(t)} \right\} = \left\{ {0_{{}} 0_{{}} 0_{{}} ...f_{1}^{(s)} f_{2}^{(s)} f_{3}^{(s)} f_{4}^{(s)} ...0_{{}} 0_{{}} 0_{{}} } \right\}$$
(2)

where in \(f_{i}^{\left( s \right)} \left( t \right),\left( {i = 1,2,3,4} \right)\) is defined as equivalent nodal forces.

$$\left\{ {f_{{}}^{(s)} (t)} \right\} = F\left\{ N \right\}$$
(3)

where F is defined as the value of the applied vertical force n on the beam, \(\left\{ N \right\}\) is defined as a function of shape and is expressed as follows:

$$\left\{ N \right\} = \left[ {N_{1} N_{2} N_{3} N_{4} } \right]^{T}$$
(4)

Shape function equations are explained by the following equations [35, 36]:

$$N_{1} = 1 - 3\, \xi^{2} + 2_{{}} \xi^{3}$$
(5)
$$N_{2} = l_{{}} \left( {\xi - 2\, \xi^{2} + \xi^{3} } \right)$$
(6)
$$N_{3} = 3_{{}} \xi^{2} - 2_{{}} \xi^{3}$$
(7)
$$N_{4} = l\left( { - \xi^{2} + \xi^{3} } \right)$$
(8)
$$\xi = \frac{x}{l}$$
(9)

Here; l is the length of the element and x is the distance to which the force F is applied in the vertical direction along with the element and Fig. 2b shows these definitions.

Considering the time step m and the time interval ∆t, the total time can be specified as follows [5]:

$$\tau = m.\Delta t$$
(10)

At any time, t = r ∆t (r = 1… m), the position of the moving force relative to the beginning of the beam is expressed as follows:

$$x_{m} (t) = v\,r\Delta t$$
(11)

Element number s is as follows, depending on the application of the moving load at any time:

$$s = Int\left[ {\frac{{x_{m} (t)}}{l}} \right] + 1$$
(12)

s − 1 and s are two nodes of the beam element. Therefore, at any time t = r Δt (r = 1 …m), the nodal forces and moments are expressed in Eqs. 13–18 if the moving force (F) acts on the s.beam element (s = 1… n). [37]:

$$F_{s - 1} = P._{{}} N_{1} = f_{1}^{(s)}$$
(13)
$$F_{s} = P._{{}} N_{3} = f_{3}^{(s)}$$
(14)
$$F_{i} = 0\left( {1\,{\text{to}}\,n,\,{\text{except}}\,s - 1\,{\text{and}}\,s} \right)$$
(15)
$$M_{s - 1} = P._{{}} N_{2} = f_{2}^{(s)}$$
(16)
$$M_{s} = P._{{}} N_{4} = f_{4}^{(s)}$$
(17)
$$M_{i} = 0\left( {1\,{\text{to}}\,n,\,{\text{except}}\,s - 1\,{\text{and}}\,s} \right)$$
(18)

where N1, N2, N3, N4 were described by previous equations between Eqs. 5–8.

Equation 9 can be rewritten according to global xm (t) instead of local x (t).

$$\xi = \frac{{x_{m} (t) - (s - 1)l}}{l}$$
(19)

2.1.1 Finite element model of the bridge for a multi trolley system

Considering the case of multiple cars, information on the dynamics of the two moving loads will be discussed in detail. If the two moving loads are assumed to both move at the velocity v and there is a certain distance between the two loads, the view of the beam under the influence of the two loads is given in Fig. 3. From the Eqs. 11 and 12 given for a beam exposed to a load to calculate the effect of the second force, the general position equation for the second load can be expressed by Eq. 20 [5]:

$$x(t) = v._{{}} r._{{}} \Delta t + x_{m} (t)$$
(20)
Fig. 3
figure 3

View of the beam subjected two moving loads

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Here, the general position of the first force xm (t) can be expressed as the distance between the two forces xf and the general position of the second force x (t). From here; The elements with a load on it can be expressed as follows:

$$s = {\text{Int}}\left[ {\frac{{v._{{}} r._{{}} \Delta t + x_{f} }}{l}} \right] + 1$$
(21)

Equation (9) can be rewritten according to global xm (t) instead of local x (t) as follows:

$$\xi = \frac{{v._{{}} r._{{}} \Delta t + x_{f} - (s - 1)_{{}} l}}{l}$$
(22)

In the same way, if the three forces are applied, the same relations can be achieved by repeating the same operations. In general, force and moment vectors can be obtained by adding the effect of two or more forces to each node together.

3 Numerical analysis method

In this study, time-varying contact forces occur on the crane wheels under operating conditions such as car moving on the crane system, speed of the load being moved, the height of the load being lifted. One of the basic principles of this study is to calculate these time-varying contact forces. When analyzing crane systems, only load and car weight are taken into consideration, and carriage and load are assumed to be symmetrical. In crane systems; dynamic effects that occur due to rope length and working speed, the car and the load on the crane system, the center of gravity of the car, and the load changes according to the working conditions on the crane system, sometimes faced with situations such as structural asymmetry. Therefore, the analyzes made under these conditions do not reflect the actual conditions.

At the same time, since crane beams have different behaviors according to the operating frequencies, it would be more appropriate to take this situation into account and to include it in the calculation. To realize this situation, firstly, crane designs were made by taking into account the actual working conditions of the crane system using a solid modeling program and by adding the rope between the load and the carriage to the system. SOLIDWORKS program was used for the solid design of crane systems.

Then the obtained solid models were transferred to the ADAMS program to analyze all effects on the beam caused by the load, rope, and trolley. For this purpose, the rope connecting the load and the trolley is modeled like a spring. In addition, the load used in the analysis (Fig. 5) was modeled exactly as the load used in the experiment (Fig. 12), and the inertia effects of the load were also taken into account. In the ADAMS program, crane beams are designed flexibly and the contact forces between each wheel and rail are calculated under different operating conditions.

In the next step, the contact forces obtained were defined in ANSYS program according to the operating conditions of the loads according to the operating speeds and the analysis was performed. Thus, the resulting contact forces include all inertia effects from the load, rope, and car.

The flow chart of this combined moving load analysis method is shown in Fig. 4.

Fig. 4
figure 4

Flow diagram of the proposed combined moving load analysis

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If the process steps in the application of this proposed analysis method are explained in detail, the crane design is modeled in real dimensions considering that the crane design will be manufactured and used in real working conditions. The elements such as motors, hooks, and reducers to be used in the crane system are determined during the design and then the same elements of the crane system are used in the manufacturing and installation. Thus, it is aimed to reduce the errors that may occur in the harmony of the results to be obtained in both analysis and experimental studies. This situation provides a more accurate approach rather than a simplified analysis. In the crane system, it is also necessary to connect the load to the carriage to take into account the effects of oscillation on the crane beams during the transport of the load. The rope acts as a spring since the carriage and the load are connected with the rope. Therefore, A spring is defined between the car and the load. A movable roller is used for the connection of load with a carriage, the load is connected to the two ropes. For this reason, the spring is defined as a single spring equivalent to two springs. In the analysis, different solid models have been created because one and more cars on the crane and different transport speeds will be carried out.

An example view of the solid models created for the investigation of the effects that occur during the transportation of the load on the carriage crane is given in Fig. 5.

Fig. 5
figure 5

An example of solid models of crane system used in ADAMS and ANSYS

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The rope used in the crane system is 8 mm in diameter and its stiffness coefficient is 400 N/mm. In these analyzes, the Coulomb friction method is used to calculate contact forces. In this method, the parameters of dynamic friction speed, dynamic friction coefficient, static friction rate, and static friction coefficient must be defined to calculate the contact forces. In the dynamic analysis, it was accepted that the wheel and the rail contacted the metal and friction in dry conditions. During the modeling, these parameters are about the selection from the program library [38]. The parameters used in the dynamic analysis are given in Table 1.

Table 1 Dynamic analysis parameters used to calculate contact forces in crane system
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Figure 6 shows the distribution of speed profiles for numerical analysis and experimental conditions according to the number of carriages and carriage movement speed in the crane system. Travel times vary depending on the bridge span and the number of carriages.

Fig. 6
figure 6

Carriages moving speed profiles

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Travel time according to the number of cars and car velocity profiles is shown in Table 2.

Table 2 Carriages travel time
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In addition to the number of cars and car speed, 12 different solid models, which were created by considering different loads, were transferred to the Adams program to analyze crane beams flexibly.

Operating speeds, friction parameters between the wheel-rail, and the definition of the spring stiffness coefficient are defined in the SOLIDWORKS program and all these adjustments are automatically transferred to the ADAMS program.

Only beams are described as flexible in the ADAMS. The crane model, in which the beams are defined as flexible, is presented in Fig. 7. After the flexible adjustment of the beams, the analyzes were performed. Four, 8, and 12 contact forces that occur between rail wheels are determined for the case of one, two, and three carriages, respectively.

Fig. 7
figure 7

View of ADAMS Model of Crane system and flexible beam model

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Although all models of the crane system under different conditions were analyzed, the results of only one of the contact forces obtained were presented. For example, the variation of the contact forces occurring on 4 wheels in a trolley crane system with a load of 1540 kg, car travel speed of 0.375 m/s, and rope length of 1 m is given in Fig. 8.

Fig. 8
figure 8

Contact forces on four wheels during the transport of a 1540 kg load with a trolley at a speed of 0.375 m/s

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When the contact forces are examined, it is understood that the change occurring in each wheel occurs at different values. It is also understood that there is excessive fluctuation in the contact forces. This situation occurs because a flexible analysis is performed. These excessive increases or decreases should be eliminated as this excessive fluctuation will cause some errors in the results obtained. these data were obtained as approximately 30 data per 1 s. This loading will increase the solution time in the finite element analysis. To compensate for this situation and extreme changes, the data in each second were averaged and these new mean values were used in the finite element analysis. Averaged wheel contact forces are shown in Fig. 9.

Fig. 9
figure 9

Change of the average of the wheel contact force for each second

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During the operation of the crane, the average contact forces occurring during the structural condition of the car, the connection of the load to the rope, and the movement of the load at a constant speed were averaged every 1 s. It is applied on areas that are as long as the road that should take in 1 s according to the speed of the car in each second as in the figure below. All forces were applied as total force as shown in Fig. 10 and dynamic analysis of the crane system was performed in case the force moves at a constant speed on the system. Also in the finite element analysis, structural damping parameters of the system have been calculated with the half-power bandwidth method [39]. For this method frequency response analysis has been conducted theoretically and experimentaly by Esim [40]. Damping ratio is calculated approximately 0.5.

Fig. 10
figure 10

View of moving total load on crane beams

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The basis of this study, which is carried out by analyzing the moving load, is to apply the external load to the finite element model based on time. For example, the application of a wheel and rail contact force is shown in Fig. 11, while the maximum load is applied in the first specified region, while in other regions it is zero. When it comes to the other area in time, the force definition in other regions is seen as zero. The identification of all contact forces according to the number of trolleys is carried out in this way.

Fig. 11
figure 11

Time-dependent application of wheel and rail contact force to the beam

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Crane systems are generally designed as a trolley. this study is focused on investigating the dynamic effects that occur on the same beam under the conditions of more than one car. For this reason, analyzes were performed on the beam according to the case of having one or more trolleys.

4 Experimental study

The deflection measurement on the beam on the crane system is an important parameter in terms of examining the dynamic effects of the system. Measuring the deflection change on the crane system is also important to investigate the efficiency of the proposed solution method theoretically. Effects on different crane system (1, 2, 3) different load transport (1540 kg, 430 kg), different car operating speeds (0.375 m/s, 0.225 m/s) and lifting height (1 m). To measure the experimental deflection measurement device was created.

In the experimental setup shown in Fig. 12, the LVDT sensor was preferred for deflection measurement. Since the crane system is on the ceiling, the connection must be made on the beam independently of the beam. Connection apparatus is prepared to connect the middle of the beam and the view showing the connections of the LVDT sensor is shown in this figure.

Fig. 12
figure 12

View of experimental crane system and LVTD sensors Connection

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Two LVTD sensors for deflection measurement were used to measure the changes in beams for a double-bridge crane system. One of the sensors is connected to the 1st beam and the other to the 2nd beam. The data acquisition mechanism and computer are shown in the picture.

The flow diagram of the device used to measure deflection over the experimental crane system is given in Fig. 13.

Fig. 13
figure 13

Block diagram of the experimental deflection measurement system

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Measurements from the experimental system were performed for the same speed, rope length, load, and run times used in numerical analysis. Linear variable differential sensors (LVDT) [41] used in the experimental system are widely used in applications such as power turbines, hydraulics, automation, aircraft satellites, nuclear reactors.

These transducers are made of 304 stainless steel which is airtight coated. it also has low hysteresis and excellent repeatability and works with DC. It can be easily used with computer-based data processors (standard) or PLCs. The sensor can be measured in one direction between 0 and 25 mm. In the process of receiving data via the sensor, data of 20 ms can be received. With this software, this period can be changed as desired. The view of the sensor and data acquisition unit used for deflection measurement is given in Fig. 12. Here, software was made to record the data and the process of receiving the data was performed.

With this software, the data received from the crane system can be monitored instantly on the computer screen.

5 Numerical analysis and experimental results

In this section, both experimental and numerical analysis results of the collapses in the beams on the crane system are presented. To calculate the changes occurring on the crane system, a mixed analysis method was applied with different analysis programs considering the inertia effects caused by the whole geometric structure of the crane system, not only as a lumped load, the effects caused by the oscillation and the modal behavior of the beam under the operating frequencies. To demonstrate the effectiveness of this proposed analysis method, the design, and analysis of the same crane system was carried out by manufacturing and installation, and experimental studies were carried out. In both experimental and simulation studies, the use of multi-carriage crane systems on how dynamic effects change and also the availability of multi-carriage crane systems under fast operating conditions compared to a one-carriage situation have been investigated. LVDT sensors are used to measure the effects on beams through a multi-car crane system. The results obtained from both methods are presented as graphs in the following figures for comparison.

While performing both experimental and experimental analyzes, studies were carried out under the same parameters that affect the crane dynamics when the car and load move together on the crane. To examine the effects of the 1st girder and the 2nd girder on the crane system, studies were carried out under different loads (430 kg, 1540 kg), a number of cars (1, 2, 3), and car operating speeds (0.225 m/s, 0.375 m/s). Rope length is taken as 1 m in numerical analysis and experimental studies.

430 kg of load, 0.225 m/s car speed, and rope length of 1 m in case of deflection change are given in Fig. 14. It is understood that the lowest load has the lowest car speed and the lowest rope length between the 1st beam and the 2nd beam and the biggest difference is in the case of the three cars.

Fig. 14
figure 14

Deflection change in the beam according to the number of cars for 430 kg load, car speed 0.225 m/s

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The deflection changes in the beams experimentally and numerically are given in Fig. 15 when the car speed is 0.375 m/s at which the transported load does not change. Similarly, with the increase in speed, the first and second beams have different dynamic behaviors. Although the difference in deflection between the beams obtained from the experimental and numerical increase in velocity is not seen at low loads, it is seen that the collapse difference between beams is decreased.

Fig. 15
figure 15

Deflection change in the beam according to the number of cars for 430 kg load, car speed 0.375 m/s

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The same work for 430 kg was carried out in the transport of 1540 kg load to determine the effects on the beam at heavy loads. The dynamic deflection change under conditions where the load is increased 1540 kg, car speed is 0.225 m/s, and rope length is considered as 1 m is presented in Fig. 16.

Fig. 16
figure 16

Deflection change in the beam according to the number of cars for 1540 kg load, car speed 0.225 m/s

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With the 1540 kg load, the dynamic effect differences in the beams became even more apparent. The maximum deflection change for the one and two-car conditions is approximately the same and the deflection difference in the first beam and the second beam is approximately the same. The effects on the third car are further increased, especially for 1 beam. In general, the deflection difference between the first and second beams in the three car cases ranges from 1 to 1.5 mm.

The deflection changes in the beams according to the number of cars during the transportation of 1540 kg load with a rope length of 1 m by increasing the car speed are given in Fig. 17.

Fig. 17
figure 17

Deflection change in the beam according to the number of cars for 1540 kg load, car speed 0.375 m/s

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When the results are analyzed, it is seen that deflection changes occur in values close to deflection amount which occurs at low speed for one car condition but the decrease in deflection values is more prominent for two and three cars. A similar change in characteristic deformation occurs at a speed of 0.225 m/s under these conditions. As the speed increases, the maximum deflection amounts decrease.

Also, the following results can be drawn when the results are examined in terms of different load, different carriage number, and different operating speed:

  1. 1.

    When the dynamic responses of the beams are evaluated, the results indicate that beams have different reactions according to the load, car operating speed, car number, and rope height and more deflection occurs in 1st beam than the 2nd beam. This may be due to the fact that there is not much force change on the second beam and the moment and inertia effects on the first beam are higher due to the effects caused by the unevenness of the center of gravity on both beams. The same effect is observed when the contact forces obtained from ADAMS are examined.

  2. 2.

    The maximum deflection occurs at the 1st beam when the rope height is 1 m at the 1st beam when the load and the car speed are maximum. Under low load and fast working conditions, deflection can be said to be close to each other.

  3. 3.

    As a result of all evaluations with respect to the number of cars, it is understood that the minimum deflection change takes place in a trolley case according to all parameters given on the crane. However, it is clear to see from graphs that deflection values that occur in two- and three-car condition decreases with the increasing car operating speed. In this case, it shows us the availability of the number of cars under the fast working conditions and to reduce the effects on the crane with multi-carriage crane systems.

6 Conclusion and discussion

Two types of numerical and experimental studies of multi-carriage crane systems were investigated with different loads, different carriage numbers, and different operating speeds. The deflection of the beams of the crane systems modeled on the number of cars with one and more are calculated with an integrated numerical solution approach.

The results obtained for both experimental and numerical studies show that the dynamic effects of the crane beams vary with the speed of movement of the load and the length of the rope to which the load is attached to the carriage. For all these reasons, the deflection values on the crane system change. With this study, it can be said that the calculations to be made statically that both beams change shape evenly in double girder bridge crane designs are insufficient. For this reason, it will be more accurate to carry out the analyzes on the crane system which takes into account the dynamic effects of the carriage speed, rope length, and car numbers. Although experimental and numerical results are obtained with similar features, there are differences in the start and finish conditions of the movement. The reason for this is that the solution interval of the contact forces obtained in the numerical analysis is chosen wide. When the solution interval is reduced, the solution time becomes too long. Another reason is the sensitivity of the sensor used in the experimental system. Considering all the structures of a crane (carriage, rope, load, solid model to be used without simplification)and under these conditions, numerical results show that the error between the experimental results is within an acceptable range. Therefore, this proposed new solution method shows that crane systems successfully capture the dynamic behavior and can be used for similar dynamic analyzes.

The authors will continue to work on the analysis of stress values and the determination of safe working conditions and safety coefficients for the system based on the results of this study.