# Efficiency of Coupled Experimental–Numerical Predictive Analyses for Inter-Story Floors Under Non-Isolated Machine-Induced Vibrations

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Research Topic

#### 1.2. Research Methods

## 2. State-of-Art on Machine-Induced Floor Vibrations

#### 2.1. Mathematical Problem

_{EQ}[31]

_{EQ}sin(wt) is the effect of the operating machinery tool, while ω represents the two possible frequency values at which the 2-DOF system would resonate, depending on the involved mass (M

_{1}, M

_{2}), stiffness (K

_{1}, K

_{2}) and damping (C

_{1}, C

_{2}) contributions. As such, a simple approach to estimate the resonance frequencies and the transmissibility magnitude for the machine-induced effects to the floor is given by

_{1}→∞), the mathematical model in Figure 2a reduces to a single DOF that transfers all the vibration source and machinery power to the structure. Depending on the damping and frequency properties of the involved systems, see Figure 2b, the machine–structure frequency ratio should be possibly limited to a maximum of 0.5, or to a value higher than 1.3, in order to minimize the dynamic amplification phenomena.

_{eff}that captures the behavior of the structure. Among others, one of its conventional definitions takes the form of

_{x}, L

_{y}the dimensions in Figure 3 and D

_{11}, D

_{22}the bending stiffnesses (per unit of width) in the x, y directions respectively (with x denoting the girders direction and y the joists).

_{t}(when relevant).

_{eff}× L

_{y}equivalent module.

#### 2.2. Reference Design Stantards

## 3. Case-Study Building

#### 3.1. Design Concept

_{x}= 67.1 × L

_{y}= 30.8 m and an aspect ratio α = L

_{x}/L

_{y}= 2.17 (Figure 5). The grid of beams and columns schematized in Figure 5 detects 6 × 2 adjacent bays (with l

_{x}= 11 × l

_{y}= 14.9 m, α = 0.74, for each one of them). Based on earlier observations and client requirements, the research study was focused on the single-span region in evidence. The structural concept, more in detail, includes a series of plinth-restrained, square columns (80 × 80 cm the cross-section) and prestressed, precast beams, that are used in the x direction to support the inter-story floor (+8 m from the foundation) and the wing-shaped members of the roof (+13 m from the foundation). All the beams have cross-section features that agree with Figure 5b, and cover a total span of 10.25 m, with 0.2 m the width of cantilever supports that are offered by the columns.

#### 3.2. Inter-Story Floor

_{cap}= 0.05 m) are kept fix.

_{0}= 0.04 m= L/365 its maximum amplitude at the mid-span section). According to the initial bow of the precast elements, the cast-in-situ concrete slab has a nominal thickness h

_{slab}comprised between 0.11 m (at mid-span) and 0.15 m (in the region of the end supports). The total mass of a typical modular unit is thus calculated in (with top slab included)

_{max}and I

_{min}are calculated for each section, while A is the cross-section and I

_{t}the moment of inertia.

#### 3.3. Materials

_{cm}for both the concrete types (up to +20% and +9% for the cast-in-situ and precast respectively, compared to their nominal grade in Table 2), and thus in a relevant modification of the actual static MoE values, given that [39]

_{ck}the characteristic compressive strength given by

#### 3.4. CNC Machines

_{OKUMA}= 7700 kg (and M

_{spindle}= 400 kg for the movable components) that roughly corresponds to ≈M

_{module}/3 (with M

_{module}given by Equation (1), with B = 2.38 m). Additional superimposed permanent loads are represented by MATSUURA [45] and BRIDGEPORT [46] machines (their weights being 4500 kg and 2700 kg) and their equipment (≈150 kg / machine).

## 4. Experimental Investigation

#### 4.1. OKUMA Machinery Center

^{2}, 1.38 m/s

^{2}and 1.6 m/s

^{2}) respectively. A variable frequency content was also observed for them, corresponding to 125 Hz, 250 Hz, and 165 Hz for W#1, W#2 and W#3. In terms of human perception, no discomfort was highlighted for these programs.

^{2}, with a frequency of 0.9 Hz. Given the severity of these accelerations and the general discomfort for the building occupants, moreover, the W#4 experimental characterization was based on acceleration records based on a limited number of cycles for each test repetition (eight in total). Figure 11b shows the synthetized signal that was experimentally derived to describe a possible continuous activity of the OKUMA.

#### 4.2. Analysis of the Inter-Story Floor

_{1,exp}= 7.4 Hz and f

_{2,exp}= 9.4 Hz respectively. Higher vibration frequencies can be also perceived in the range from ≈12 Hz to ≈40 Hz.

^{2}.

#### 4.3. Damping

_{exp}≈ 7.5% (and a maximum of ξ

_{exp}≈ 9%) for the floor under the W#4 program (s#4 data). A close correlation was found with the measured damping in s#4, with the machines at rest (W#0 setup). Close to the OKUMA footprint, finally, the calculated damping resulted in a mean ξ

_{exp}≈ 4–5% value (W#4, s#5 control point).

## 5. Numerical Analysis

_{x}× l

_{y}its size), that was preliminary validated in its basic input features, under static loading conditions.

#### 5.1. Description of Floor and Machines

_{spindle}was used to account for the spindle movements. The experimental history of machinery accelerations (i.e., Figure 9 and Figure 11) was assigned to the M

_{spindle}lumped mass, and thus transferred to the floor. In doing so, a special attention was paid for the schematization of the machinery effects. A rigid link was first introduced to connect M

_{spindle}with the floor. To this aim, an intermediate ‘RP’ node in Figure 16c was positioned in the center of mass of the OKUMA footprint. This RP was set as reference node for an additional ‘coupling’ kinematic constraint, that was used to connect the RP node (and thus M

_{spindle}) to a set of four base rigid links, that were used to schematically describe the OKUMA foundation restraints (washers) in Figure 8a and to transfer the machine acceleration to the slab.

_{1}= ∞) for the undamped OKUMA machine (C

_{1}= 0). On the other side, the inter-story floor was numerically described with detailed geometrical and mechanical properties, so as to capture its stiffness K

_{2}, mass M

_{2}, and damping C

_{2}. Following Section 4.3, all the dissipative capacities were lumped on the floor slab (uniform modal damping).

#### 5.2. Static Analysis

_{OKUMA}is in fact calculated from the acceleration peaks in Figure 9 and Figure 11 and the available mass contributions for the machine, the corresponding mid-span deflection can be calculated with the available tools.

_{comp}the bending stiffness of the composite resisting section, as obtained from Table 1. Both Equation (8) and the corresponding numerical analysis resulted in a maximum static deflection in the order of 0.029 mm (W#4 process).

#### 5.3. Natural Frequency Results

^{3}). As far the slab in the top is fully disregarded (in the same way of the supported CNC machines and equipment), and the mean dynamic MoE is derived from Equation (7), Equation (10) would result in a fundamental frequency f

_{1,an}= 7.79 Hz that roughly capture the experimental prediction from Figure 18.

_{1,exp}of the floor, the same model still suggests a prevailing ‘joist panel mode’ for the structure object of study. Moreover, the same prediction f

_{1,an}is still lower than 10 Hz (Figure 4) and thus would suggest the need of dedicated early-stage design considerations.

_{y}its size), is representative of a single double tee module (B = 2.38 m), with the top slab and the supported OKUMA + equipment masses. The major limit of the ‘one-module’ model is that the adjacent modules are disregarded, in the same way of the flexibility of the end restraints. Ideal simply supports are in fact used to restraint the webs, in place of the actual precast beams.

_{x}× 2l

_{y}its size) and the slab continuity in both the x and y directions, thus resulting in 62,000 elements and 390,000 DOFs.

_{1,empty}= 7.81 Hz and f

_{2,empty}= 9.98 Hz for the empty floor, based on the one-bay model predictions), thus resulting in even more pronounced serviceability issues (i.e., increased flexibility, and thus sensitivity to vibrations).

## 6. Coupled Experimental–Numerical Vibration Assessment

#### 6.1. Experimental Configuration

_{exp}= 9% was taken into account for the analyses, based on the experimental feedback (Section 4.3).

_{1}−f

_{2}natural frequencies. Moreover, the s#4 control point and the second mode of vibration of the structure were usually found to be associated with more pronounced dynamic effects (i.e., acceleration peaks), compared to the central s#1 point, as an effect of the machine-induced vibrations and the related torsional deformations of the deck.

^{2}. For the same W#4 loading scheme, moreover, non-null vertical acceleration records were numerically predicted for the control points s#2 and s#3, as a direct effect of the intrinsic flexibility of the supporting precast beams, thus confirming the potential (compared to ideal rigid boundaries) of refined FE models for vibration assessment purposes.

#### 6.2. Design Configuration

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**General view of the floor object of investigation, with evidence of CNC machines and equipment.

**Figure 2.**Reference methods for the analysis of machine–structure interaction phenomena: (

**a**) 2-DOF model and (

**b**) transmissibility of the machine force.

**Figure 3.**Efficient width approach for the analysis of long floors. Adapted from [32] under the terms and condition of CC BY-NC 4.0 license.

**Figure 4.**Velocity peak limits to prevent structural damage in industrial buildings exposed to short-term vibrations.

**Figure 5.**Case-study building: (

**a**) plan view of the inter-story floor, with (

**b**) cross-sectional details of the precast concrete beams and columns (nominal dimensions in m).

**Figure 6.**Reference modular unit for the inter-story floor under investigation: (

**a**) transversal cross-section and (

**b**) plan/side views of a single module (nominal dimensions in m).

**Figure 7.**Plan view of the CNC machines on the floor region object of study (nominal dimensions in m), with details of the OKUMA machinery center.

**Figure 8.**OKUMA machinery center: (

**a**) detail of the typical base restraint (foundation washer) and (

**b**) its spindle (in evidence, the mp#1 accelerometer), with (

**c**) example of video-tracking acquisition of the vertical displacements.

**Figure 9.**Experimental records of the spindle vertical acceleration from three different working activities of the OKUMA machinery center (mp#1): (

**a**) W#1, (

**b**) W#2, and (

**c**) W#3.

**Figure 10.**W#3 process for the OKUMA machinery center (mp#1): (

**a**) experimental record of the spindle vertical acceleration, with (

**b**) detail view.

**Figure 11.**W#4 process for the OKUMA machinery center (mp#1): (

**a**) experimental record of the spindle vertical acceleration (single cycle) and (

**b**) derivation of the corresponding synthetized signal for a continuous activity.

**Figure 12.**Reference setup for the field dynamic experiments: (

**a**) plan view, with dimensions in m; (

**b**) detail of mp#2 (base of the machine frame); and (

**c**) global view of the examined floor region (in evidence with red circles, the position of sensors s#4, s#5, and mp#2).

**Figure 13.**Experimental records for the inter-story floor region with the OKUMA at rest (W#0). Measured (

**a**) vertical acceleration and (

**b**) corresponding power spectral density (PSD). In evidence, a selection of 5 s of acquisition (data from sensor s#4).

**Figure 14.**Experimental PSD for the examined floor region (W#4), as obtained from (

**a**) mp#2 (base of the machine frame), (

**b**) s#5 (floor), and (

**c**) s#4 records (floor mid-span). In evidence, a selection of 5 s of acquisition.

**Figure 15.**Experimental acceleration for the examined floor region (W#4), as obtained from s#4 records (floor mid-span). In evidence, a selection of 5 s of acquisition.

**Figure 16.**Modeling approach for the FE model of the inter-story floor region (ABAQUS, one-bay FE model): (

**a**) cross-section and (

**b**) side view details for a modular unit, with (

**c**) final assembly (dimensions in m).

**Figure 17.**Deflection of the floor under the equivalent static force peak of the OKUMA machine (ABAQUS, one-bay FE model, W#4, legend values in m, scale factor: ×50,000).

**Figure 18.**Numerical vibration modes of the inter-story floor region with all the CNC machines at rest (ABAQUS ‘frequency’ analysis, one-bay FE model). (

**a**) f

_{1}= 7.37 Hz. (

**b**) f

_{2}= 9.41 Hz

**Figure 19.**Fundamental natural vibration modes with all the CNC machines at rest (ABAQUS ‘frequency’ analysis, 2 × 2-bay model): (

**a**) f

_{1}= 7.38 Hz, (

**b**) f

_{2}= 9.42 Hz, and (

**c**) intermediate modes.

**Figure 20.**SSD analysis of the inter-story floor region with the working OKUMA (ABAQUS, one-bay FE model, W#4). Evolution of (

**a**) vertical acceleration peaks in the s#1 and s#4 control points, with (

**b**) comparison of absolute acceleration peaks experimental configuration (synthetized signal, experimental dynamic MoE for concrete and ξ

_{exp}= 9%).

**Figure 21.**SSD analysis of the inter-story floor region with the working OKUMA (ABAQUS, one-bay FE model, W#2). Evolution of vertical velocity peaks in the s#1 and s#4 control points. Experimental configuration (synthetized signal, experimental dynamic MoE for concrete and ξ

_{exp}= 9%).

**Figure 22.**SSD analysis of the inter-story floor region with the working OKUMA (ABAQUS, one-bay FE model): (

**a**) vertical acceleration, (

**b**) velocity, and (

**c**) displacement peaks in the s#1 and s#4 control points. Experimental configuration (synthetized signal, experimental dynamic MoE for concrete and experimental damping ξ

_{exp}= 9%).

**Figure 23.**SSD analysis of the inter-story floor region with the working OKUMA (ABAQUS, one-bay FE model): (

**a**) vertical acceleration, (

**b**) velocity, and (

**c**) displacement peaks in the s#1 and s#4 control points. Design configuration, with synthetized signal, nominal dynamic MoE for concrete and conventional damping ξ = 3%).

**Figure 24.**SSD analysis of the inter-story floor region with the working OKUMA (ABAQUS, one-bay FE model, W#4): (

**a**) vertical acceleration, (

**b**) velocity, and (

**c**) displacement peaks in the s#1 and s#4 control points, as a function of the dynamic MoE of concrete and damping.

Element | Section Parameter | |||
---|---|---|---|---|

A (m^{2}) | I_{max} (m^{4}) | I_{min} (m^{4}) | I_{t} (m^{4}) | |

Slab (max, B = 2.5 m) | 0.373 | 0.1953 | 0.0070 | 0.0027 |

Slab (min, B = 2.5 m) | 0.275 | 0.1432 | 0.0027 | 0.0010 |

Double tee module (B = 2.5 m) | 0.392 | 0.1578 | 0.0254 | 0.0017 |

Beam #1 | 0.076 | 0.0645 | 0.0182 | 0.0373 |

Beam #2 | 0.061 | 0.0943 | 0.0326 | 0.0507 |

**Table 2.**Reference mechanical properties for the concrete types in use (nominal and experimental values)

Mix | Element | Nominal | Experimental (avg.) | ||
---|---|---|---|---|---|

Grade | E_{cm} (MPa) | f_{ck} (MPa) | E_{cm} (MPa) | ||

Cast-in-situ concrete | Continuous slab | C25/30 | 31,476 | 53.3 | 37,893 |

Precast concrete | Columns, beams, double tee floor modular units | C50/67 | 38,214 | 76.1 | 41,674 |

**Table 3.**Experimental investigation of the OKUMA machine (vertical acceleration programs, mp#1 sensor).

Spindle Analysis (mp#1 Sensor) | |||
---|---|---|---|

W#n | Records | Acceleration Peak (m/s^{2}) | Frequency (Hz) |

1 | 3 | 0.6 | 125 |

2 | 3 | 1.38 | 250 |

3 | 3 | 1.6 | 165 |

4 | 9 | 4.95 | 0.9 |

**Table 4.**Experimental investigation of the floor under the OKUMA activity (vertical acceleration programs, various acquisition sensors)

W#n | Floor Analysis | ||
---|---|---|---|

Sensor | Records | Acceleration Peak (m/s^{2}) | |

0 * | s#4 | 5 | 0.26 |

4 | mp#2 | 3 | 0.52 |

4 | s#4 | 3 | 0.28 |

4 | s#5 | 3 | 0.21 |

**Table 5.**Comparison of natural vibration frequencies for the inter-story floor with non-operating CNC machines, based on different modeling approaches

FE model | |||||
---|---|---|---|---|---|

Experimental | Analytical (Equation (10)) | 1-Module | One-Bay | 2 × 2-Bay | |

n | f (Hz) | f (Hz) | f (Hz) | f (Hz) | f (Hz) |

1 | 7.40 | 7.79 | 7.39 | 7.37 | 7.38 |

- | - | - | - | - | 7.61 |

- | - | - | - | - | 8.17 |

2 | 9.40 | - | - | 9.41 | 9.42 |

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**MDPI and ACS Style**

Bergamo, E.; Fasan, M.; Bedon, C. Efficiency of Coupled Experimental–Numerical Predictive Analyses for Inter-Story Floors Under Non-Isolated Machine-Induced Vibrations. *Actuators* **2020**, *9*, 87.
https://doi.org/10.3390/act9030087

**AMA Style**

Bergamo E, Fasan M, Bedon C. Efficiency of Coupled Experimental–Numerical Predictive Analyses for Inter-Story Floors Under Non-Isolated Machine-Induced Vibrations. *Actuators*. 2020; 9(3):87.
https://doi.org/10.3390/act9030087

**Chicago/Turabian Style**

Bergamo, Enrico, Marco Fasan, and Chiara Bedon. 2020. "Efficiency of Coupled Experimental–Numerical Predictive Analyses for Inter-Story Floors Under Non-Isolated Machine-Induced Vibrations" *Actuators* 9, no. 3: 87.
https://doi.org/10.3390/act9030087