Dynamic Response of Beams Under Random Loads

Abstract

In engineering, the study of the dynamic response of structures subjected to non-deterministically variable loads is particularly important, especially when considering the damage that such loads can cause due to fatigue phenomena. This is the case, for example, of the vibrations that a satellite must withstand during the launch phase. In the preliminary design phases, it is very useful to have semi-analytical calculation methodologies that are sufficiently reliable but, at the same time, simple. In the technical literature, there are numerous publications that deal with the study of the random dynamic response of beam models. In general, the presented studies are rather complex, and the dynamic solutions are often obtained in the time domain. The case of a linear elastic uniform cantilever beam model is considered here, for which the analytical expressions of the transfer functions for acceleration, displacement, bending moment, and bending stress are calculated, taking as input the acceleration assigned to the root section or an external lateral load. Knowing the spectral density of the input loads, the spectral densities of all the above-mentioned variables are calculated along the beam axis, assuming stationary and ergodic random processes. Using the spectral density of each output variable, the effective value (RMS) is obtained via integration, which allows for a preliminary estimate of the severity of the working conditions of the beam. The spectral density of the responses also allows us to quickly highlight the contribution of each natural vibration mode as the spectrum of the load varies. The results were obtained using simple spreadsheets available to the reader.

1. Introduction

In the technical–scientific literature, many articles and books have been published that are dedicated to the study of the dynamic response of structures subjected to non-deterministic loading conditions [1] (pp. 436–517) and [2] (pp. 408–476). Some of the proposed solutions, presented in analytical form, concern beam models [3,4,5,6,7] that are very well suited to the preliminary study of the dynamics of road and railway bridges [8] and blades of large wind turbines or aircraft wings [9] (pp. 421–429). More recently, thanks to the systematic use of the finite element method, numerical solutions have been proposed for more geometrically complex structural configurations. These solutions allow for a more faithful representation of both the structural deformations and the stress states that are produced due to random loads, loads that, over time—even if not particularly high—can produce insidious and dangerous fatigue damage [10,11,12,13,14,15].

As mentioned, a sector particularly affected by these issues is bridge design and construction. Bridge structures are subject to external loads due to the weight and movement of vehicles crossing the bridges themselves, the effects of temperature changes, the non-stationary effects of aerodynamic loads induced by the wind, especially on particularly flexible supporting structures, dynamic loads produced by earthquakes, and possible effects of geometric imperfections or misalignments that may occur during the construction phase [8,10,11,12,16,17,18,19,20].

Another sector of civil construction that is sensitive to these problems is that of increasingly tall and slender skyscrapers, for which the dynamic stresses induced by vibrations caused by the action of wind can be particularly strong [21,22].

Even if extensive use is made of structural models and highly sophisticated numerical calculation codes in the preliminary design phase, in all sectors of industrial engineering (from civil to aerospace), for designers, the availability of analytical or semi-analytical solutions (even if simplified) that provide rapid and clear information on the dynamic response of structures subjected to a generic non-deterministic load condition still has its own technical validity, especially if the way in which the solutions are used is formally simple, easily understandable, and easily interpretable.

In civil or mechanical engineering, the analysis of stresses produced by randomly variable loads allows for a more efficient design of load-bearing structures: that is, both the static strength characteristics and the properties of the dynamic response of the structures are considered fundamental. In some cases, by observing the degradation of these latter properties over time, it is possible to trace the presence, distribution, and extent of damage or cracking of the structures themselves [23,24].

In this regard, techniques based on the use of artificial intelligence are being developed that allow us to both diagnose the presence of possible damage and predict its distribution and criticality. An example of the application of machine learning techniques to the aerostructures sector has been presented in [25].

In recent decades, the installation of wind generators of increasing power and size in the renewable energy sector has led to an in-depth statistical study of both the wind conditions of the installation sites and the durability of the generators themselves (of the moving mechanical parts and the blade structures), subject to load conditions that are highly variable over time and, above all, intrinsically non-deterministic. Wind turbine blades can be modeled, as a first approximation, as if they were beams: cantilevered in the case of horizontal-axis wind turbines (HAWTs) or constrained along the span in one or more distinct sections in the case of vertical-axis wind turbines (VAWTs) [26,27,28,29,30,31,32].

In the field of aerospace engineering, the analysis of deterministic and non-deterministic vibrational environments is fundamental both for the design and experimental verification of aerostructures components or entire space systems [33] (pp. 401–405), [34,35,36,37].

More specifically, in aeronautical engineering, the definition of random vibration environments and the determination of dynamic responses in stochastic form concerns phases of the operational life of an aircraft both on the ground (taxiing) [2] (pp. 447–451) and in flight [9] (pp. 416–419). A further source of non-deterministic loads is related to aeroacoustic phenomena, which are very important for both aeronautical and space engineering. These loads, in addition to producing high-intensity random vibrations, can cause fatigue failures of thin-walled structures that are particularly sensitive to the action of randomly oscillating pressures. A very first example of calculation of the effects of aeroacoustic environments, which has a historical value if we want, is reported in [5]. In this work, the effects of aeroacoustic pressures on the response of aluminum panels for aeronautical use are first discussed; secondly, a mathematical model of a beam is developed to preliminarily quantify the effects of the damping parameters on the magnitude of the response. In this case, by applying Galerkin’s method, a single modal equation is obtained and solved, assuming that the fundamental mode is predominant. To obtain the Gaussian response parameters, acoustic pressure excitation is assumed to be stationary Gaussian and ergodic with a zero mean.

In addition to the phenomena already mentioned, linked to gust-induced loads, turbulent boundary layers, and aeroacoustics, it is also necessary to consider flight conditions that can become globally critical. In fact, non-deterministic vibrational phenomena that occur in the pressure field surrounding the aircraft’s lifting surfaces in all flight regimes, from subsonic to supersonic, can trigger unstable aeroelastic responses (flutter or flutter-buffet) [9] (pp. 421–429), [38,39,40,41].

In space engineering, high dynamic stresses occur, especially during the launch phase or during the re-entry phase of reusable parts of the launcher [36]. Stresses depend on vibrations produced by the motion of the rotating parts of liquid-propellant space engines, aeroacoustic pressure waves that hit the external walls of the launcher, pressure waves produced by the high-speed and high-energy exhaust of hot gases exiting the nozzles, and pressure oscillations that develop on the external walls of a launcher during the transonic flight phase when the aerodynamic flow is generally highly unstable and characterized by a complex system of shock waves [33] (p. 47), [40,41].

In bridge engineering, the non-deterministic characteristics of loads are linked to the number, size, mass, and movement of vehicles and/or trains that cross the bridges themselves. In fact, the number, size, mass, and speed of vehicles that will cross a newly built bridge daily cannot, in principle, be defined in a deterministic manner and, therefore, the analysis of the stresses—and, consequently, the development of the structural project itself—can only be conducted by following a statistical approach [8,11,12,17].

As the time history of stresses depends on the dynamic response of the structures, it is of fundamental importance to have calculation methodologies that allow for the flexible, clear, and sufficiently reliable prediction of the local stress state on which, as is known, the durability characteristics of the structure also depend.

From a purely methodological point of view, two types of dynamic load can be distinguished: (a) a dynamic load that acts directly on the structures, for example, a pressure acting on a panel or a lateral load applied to a beam, and (b) an indirect dynamic action due to the movement of the structure’s supports (base-driven problem). In space engineering, an example of the first case is the aero-acoustic pressure field that directly acts on the surface of a solar panel of a satellite stowed inside the fairing of a launcher during atmospheric flight phases. An example of the second case is given by the deterministic and non-deterministic vibrations that are transmitted to a sensor or an antenna, especially during the launch phases, through their support structures mechanically connected to the primary structure of a satellite system. In some cases, this second type of dynamic load (especially in the case of bridges or aerospace structures) can be exploited for the production and storage of electrical energy for the on-site power supply of structural monitoring systems, based, for example, on the use of piezoelectric materials or transducers such as those studied in [42,43,44,45,46].

In order to develop realistic engineering applications, it is necessary to have dynamic solutions that consider load conditions whose frequency spectra have a generic shape.

As an example, the celebrated work in [3] calculated the response of a uniform beam to stationary random excitation, considering different damping mechanisms, subjected to a distributed transverse loading process (uncorrelated spacewise) described by either spectral densities of ideal white noise or band-limited white noise. This work, although very elegant in its mathematical formulation, does not consider the case of external loads with a non-uniform spectral density profile. Similarly, the work outlined in [4], based on the application of the modal formulation of the dynamic response of simple beams, defines the solution under the action of stationary random excitations perfectly correlated in the space and frequency domains, which can, therefore, be represented by a constant value of the spectral density.

In [6,7], closed-form solutions for the dynamic responses are found for uniform, simply supported beams subjected to a stationary excitation that, again, is white both in space and time (the so-called rain-on-the-roof excitation).

In the already cited book [33], the calculation procedure is briefly presented, and the results relating to the spectral density of the acceleration response of a rectangular sandwich plate (the acceleration response calculated at the center of the plate) are discussed. The plate, simply supported on the edges, is stressed by a random pressure distribution whose spectral density is known (the spatial correlation is managed by dividing the calculation domain into autocorrelated subregions).

Within the broad engineering panorama described (albeit in a synthetic manner), this paper presents, by way of example, a methodology for simple and direct calculation of the statistical properties of the dynamic response of linear elastic beams or linear elastic beam systems, assuming that their modal characteristics are known and assuming that the external loads applied to the system are defined by means of spectral densities. The methodology formally follows the approach described in [33], but the application is specifically developed for the technical beam model. This is what is normally performed, for example, in the space engineering sector when—by adopting certainly more sophisticated calculation methods and having selected a specific launcher among those existing and available on the market—structural engineers want to preliminarily define the characteristics of the dynamic response of satellites or their subsystems, modeled as a first approximation as beams, under the action of non-deterministic vibrational spectra established in the launcher’s user manual.

2. Materials and Methods

Although the direct solution of the equilibrium equations of dynamics is often used in the literature for beam models, when load conditions can be described mathematically in an explicit manner [3,6,47,48,49,50], from a technical point of view, for structures of various types and therefore also for beam-like structures, the modal superposition method is preferred for the analysis of random vibrations involving several modes of vibration for both linear and nonlinear formulations [4,8,51,52,53,54]. In such a case, the solution procedure presupposes (a) the modal analysis (possibly, where necessary, also including free body effects), (b) the definition and description of the fluctuating load conditions, (c) the solution process, and (d) the output of the structural response results. Even under simplified assumptions, the methods used to obtain a random solution often require that the space–time correlation function for the external loads be known, but this is generally not sufficiently emphasized. Knowing an analytical expression of the load correlation function in real cases is not at all simple or obvious. In other cases, the spectral density of the loads is assigned but simply expressed as white Gaussian noise. In this section, all the abovementioned steps (a) to (d) are illustrated for some cases concerning uniform beams. The dynamic equilibrium equations refer to a “classical” model of a straight, uniform, and slender beam in which the deformation effects induced by shear are neglected, with free-to-warp sections and with homogeneous and isotropic material. Therefore, to obtain the solutions of the random dynamic responses, the so-called Euler–Bernoulli beam model is used, assuming that a generic distribution of the spectral density of the random loads is known.

2.1. Beam Dynamics Equations

Lagrange’s method [2] (pp. 253–260) is particularly efficient for writing the equations of motion of systems with multiple degrees of freedom (whether discrete or continuous). From a mathematical point of view, beams are continuous systems, and therefore, it is convenient to represent the deformation characteristics of these objects using their own vibration modes.

In this way, mathematically, the degrees of freedom of a beam become its own modes of vibration. Therefore, the equations of motion should be written with reference to them. In this sense, the energetic approach (Lagrange’s method) is very useful: it is sufficient to appropriately write the expressions of the kinetic energy, the elastic potential energy, and the work of the external forces as a function of appropriate variables (the so-called Lagrangian variables). For a cantilever beam problem (Figure 1), assuming from the beginning the natural modes of vibration as degrees of freedom, the Lagrangian variables are nothing other than the modal coordinates that weigh the contribution of the individual modes, as described by the following formula (considering only the first $N$ modes of vibration of a straight beam):

$$v\left(x,t\right)=\sum _{i=1}^N{\varphi }_i\left(x\right)·q_i\left(t\right),$$

(1)

where $v\left(x,t\right)$ represents the lateral displacement of a generic section along the beam axis ($x$ is the section’s abscissa and $t$ is the time variable), ${\varphi }_i\left(x\right)$ is the generic mode of vibration of the beam, and $q_i\left(t\right)$ is the generic modal coordinate. The mathematical expressions of the modes depend on the boundary conditions of the problem (as well as on the physical characteristics of the beam).

The modal coordinate $q_i\left(t\right)$ depends on both the initial conditions and the loading conditions applied to the beam: concentrated loads and/or distributed loads; displacements, velocities, and accelerations imposed on specific sections of the beam; concentrated and/or diffuse temperature variations, etc. Moreover, these loading conditions can be deterministic or non-deterministic.

Lagrange’s equation for a generic modal coordinate is given by the formula below:

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial T}{\partial {\dot{q}}_i}}\right)+{\displaystyle \frac{\partial U^e}{\partial q_i}}={\displaystyle \frac{\partial L^{ext}}{\partial q_i}} , i=1,N .$$

(2)

In Equation (2), $T$ is the kinetic energy of the system (which is assumed here to depend only on the velocity components), $U^e$ is the elastic potential energy (strain energy) of the system, $L^{ext}$ is the work performed by external forces, and ${\dot{q}}_i$ is the time derivative of $q_i$. To formally obtain the equations of motion, we will neglect for simplicity the effects of the structural damping (which, as we will see, can be inserted by considering an additional term in the final equations). From a practical point of view, the damping forces can be considered as external forces and therefore incorporated into $L^{ext}$. The next step is to express the various quantities as functions of $q_i$ and ${\dot{q}}_i$ and then calculate the derivatives.

Referring, for example to a cantilever beam of length $L$ subjected to a distributed load $p\left(x,t\right)$ and to a concentrated force $P\left(t\right)$ applied in a generic section $x_P$ (Figure 2), independently of the constraint conditions of the extreme sections of the beam, the details of these calculations are summarized in Appendix A.

Considering a typical loading condition of vibratory environments (both deterministic and nondeterministic), i.e., the so-called base-driven problem, the input can be represented, for example, by the acceleration ${\ddot{v}}_B\left(t\right)$ applied to the constrained section of the beam. This is the classic model that describes the effects of the actions induced on a civil structure by earthquakes [1] (pp. 544–610), [8,18] or the effects of strong vibrations that stress a satellite and its subsystems, especially during the launch phase [33] (pp. 38–51).

Considering a cantilever beam model (Figure 3), the absolute lateral displacement along the beam $v_{Tot}\left(x,t\right)$ can be written as follows:

$$v_{Tot}\left(x,t\right)=v_B\left(t\right)+\sum _{i=1}^N{\varphi }_i\left(x\right)·q_i\left(t\right),$$

(3)

where $v_B\left(t\right)$ represents the lateral displacement of the clamped section of the beam (the base), ${\varphi }_i\left(x\right)$ is the generic natural mode of vibration of the clamped beam, $q_i\left(t\right)$ is the generic modal coordinate, and $N$ is the number of natural modes involved in the analysis.

The equations of motion will now contain the effects of the acceleration of the base ${\ddot{v}}_B\left(t\right)$ (typically vibratory environments are described in terms of acceleration), which effectively becomes the external load of our structural model. The details of these calculations are summarized in Appendix B.

In both Appendix A and Appendix B, it is shown how, regardless of loading conditions, when adopting the modal representation technique of the dynamic response of a structure, for each degree of freedom considered, the equation of motion has the following form (also including the effects of damping) [33] (p. 197):

$$m_{gi}·{\ddot{q}}_i\left(t\right)+2·m_{gi}·{\omega }_{ni}·{\zeta }_i·{\dot{q}}_i\left(t\right)+k_{gi}·q_i\left(t\right)=f_{gi}\left(t\right),$$

(4)

the terms are generalized mass $m_{gi}$, generalized stiffness $k_{gi}$, natural circular frequency ${\omega }_{ni}$, damping ratio ${\zeta }_i$, and generalized force $f_{gi}\left(t\right)$.

2.2. Solution of Beam Dynamics Equations

Equation (4), which describes the evolution of the generic modal coordinate in the time domain, can be solved directly with different analytical methodologies depending on the deterministic mathematical form of $f_{gi}\left(t\right)$.

In this case, the part of the time solution dependent on external forces (the steady state solution), $q_{fi}\left(t\right)$, can be calculated using Duhamel’s integral [33] (pp. 95–108):

$$q_{fi}\left(t\right)={\displaystyle \frac{1}{m_{gi}·{\omega }_{di}}}·{\int }_0^t{f}_{gi}\left(\tau \right)·e^{-{\zeta }_i·{\omega }_{ni}·\left(t-\tau \right)}·\mathit{sin}\left[{\omega }_{di}·\left(t-\tau \right)\right]·d\tau , { \omega }_{di}={\omega }_{ni}·\sqrt{ 1-{\zeta }_i^2 }.$$

(5)

On the other hand, this cannot be achieved in the case where the loading condition is defined in a statistical way (i.e., it is of a random type). In this case, it is generally convenient to transform the equations and solve the problem in the frequency domain. To this end, we can us the Fourier transform ($j$ being the imaginary unit and $\omega =2·\pi ·f$):

$$Q_i\left(f\right)={\int }_{-\infty }^{\infty }{q}_i\left(t\right)·e^{-j·2·\pi ·f·t}·dt ,$$

(6)

and remember the relationships between the derivatives of the transforms:

$${\dot{Q}}_i=j·\omega ·Q_i \mathrm{a}\mathrm{n}\mathrm{d} {\ddot{Q}}_i=j·\omega ·{\dot{Q}}_i ,$$

(7)

we can, for example, rewrite Equation (4) in terms of ${\ddot{Q}}_i$, the Fourier transform of the acceleration of the i-th modal coordinate:

$$Q_i=-{\displaystyle \frac{{\ddot{Q}}_i}{{\left(2·\pi ·f\right)}^2}} \mathrm{a}\mathrm{n}\mathrm{d} {\dot{Q}}_i=-j·{\displaystyle \frac{{\ddot{Q}}_i}{2·\pi ·f}} .$$

(8)

After some simple mathematical steps, we achieve the following:

$${\ddot{Q}}_i\left(f\right)·\left[ 1{-\left( {\displaystyle \frac{f}{f_{ni}}} \right)}^2+j·2·{\zeta }_i·\left( {\displaystyle \frac{f}{f_{ni}}} \right) \right]=-{\left( {\displaystyle \frac{f}{f_{ni}}} \right)}^2· {\displaystyle \frac{F_{gi}\left(f\right)}{m_{gi}}} ,$$

(9)

where $F_{gi}\left(f\right)$ is the Fourier transform of $f_{gi}\left(t\right)$. Finally,

$${\ddot{Q}}_i\left(f\right)=H_i\left(f\right)·F_{gi}\left(f\right) ,$$

(10)

with

$$H_i\left(f\right)={\displaystyle \frac{1}{m_{gi}}}·{\displaystyle \frac{-{\left( \frac{f}{f_{ni}} \right)}^2}{\left[ 1{-\left( \frac{f}{f_{ni}} \right)}^2+j·2·{\zeta }_i·\left( \frac{f}{f_{ni}} \right) \right]}} .$$

(11)

$H_i\left(f\right)$ is the transfer function for the generic modal coordinate, which correlates, in this case, the acceleration response to the input. It is a complex function of frequency. The transfer function depends on the geometric and mechanical characteristics of the structural model. For a beam model, the input can consist of a concentrated force and/or a distributed force (see Appendix A). Considering a base-driven system, Equation (10) gives the Fourier transform of the i-th modal acceleration response relative to the base (see Appendix B). For the cases discussed here, the Fourier transform $F_{gi}\left(f\right)$ depends on the transforms of the functions $p\left(x,t\right)$, $P\left(t\right)$, and ${\ddot{v}}_B\left(t\right)$.

By representing quantities of interest in the frequency domain, it is possible to evaluate a random forcing environment or a random response in terms of the mean square response thanks to Parseval’s theorem [33] (p. 121). Assuming as valid the classical hypotheses that define a random process as stationary and ergodic [2] (p. 412), the mean square value of a function $z\left(t\right)$, defined in the time domain, calculated over a certain time interval $T$, can be obtained by integrating its spectral density in the frequency domain or its power spectral density (PSD) $W_z\left(f\right)$, which depends substantially on the square of the modulus of its Fourier transform $Z\left(f\right)$, according to the mathematical definition of auto-PSD in a stationary ergodic random process.

In the literature, random external loads are often defined in a mathematically simple way as space- and time-wise ideal white noise to allow closed-form analysis [1] (pp. 611–616), [6,7,48,52]. This technique is, however, quite laborious and not very flexible.

In this work, it is assumed that the “random processes” that define the external loads and possess properties of statistical regularity [2] (p. 409) are described directly by their spectral densities. Moreover, for lightly damped systems, as typically occurs in structural engineering, since the response processes produced by different modes of vibration are almost statistically independent, the cross-spectral density functions for different modal responses are almost equal to zero [1] (pp. 508–511); therefore, to produce technically acceptable data, it is sufficient to calculate only the auto-PSDs of the response parameters.

Based on these considerations, the calculation of random beam responses described here depends exclusively on the knowledge of the auto-PSD of the forcing function; hence, the mean square value of $z\left(t\right)$ can be calculated using the following formula:

$${\left( \overline{z\left(t\right)} \right)}^2={\int }_0^{\infty }{W}_z\left(f\right)·df .$$

(12)

Now, given the frequency response law, it is possible to determine its PSD as well, provided that the PSD of the input is known. For example, referring to Equation (10), we have the following ($W$ symbols represent the PSDs and $‖H_i\left(f\right)‖$ represents the module of $H_i\left(f\right)$; see, as a comparison, Equation (41) in [3]):

$$W_{{\ddot{Q}}_i}\left(f\right)={‖H_i\left(f\right)‖}^2·W_{F_{gi}}\left(f\right) .$$

(13)

Then, from (a) the modal analysis of the system, (b) the transfer functions referring to the modal coordinates, and (c) the knowledge of the spectral density of the input, it is possible to calculate the spectral density of the response for the i-th modal coordinate using Equation (13). By superimposing the effects, using Equation (1) or Equation (3) as appropriate, it is possible to calculate the spectral density of the response by referring to the physical quantity of interest (displacement, velocity, or acceleration of a section of the beam).

The square root of Equation (12) provides the so-called RMS (root mean square), which, in the case of a purely random (zero mean value) Gaussian phenomenon, corresponds to the standard deviation of the statistical distribution of the variable under examination.

2.2.1. Uniform Cantilever Beam Subjected to Distributed and Concentrated Loads

In this case, to simplify the problem, let us imagine that $p\left(x,t\right)=r\left(x\right)·G\left(t\right)$, if $\widehat{G}\left(f\right)$ and $\widehat{P}\left(f\right)$ indicate the Fourier transforms of $G\left(t\right)$ and $P\left(t\right),$ respectively (for the definition of $f_{gi}\left(t\right)$, see Equation (A12)):

$$F_{gi}(f)=\widehat{G}\left(f\right)·\left[ {\int }_0^{L}r\left(x\right)·{\varphi }_i\left(x\right)·dx \right]+\widehat{P}\left(f\right)·{\varphi }_i\left(x_P\right) ,$$

(14)

therefore, the transfer function of the acceleration response is calculated using two terms:

$$H_G\left(x,f\right)=\sum _{i=1}^N{\varphi }_i\left(x\right)·H_i\left(f\right)·{\int }_0^{L}r\left(x\right)·{\varphi }_i\left(x\right)·dx \mathrm{a}\mathrm{n}\mathrm{d} H_P\left(x,f\right)=\sum _{i=1}^N{\varphi }_i\left(x\right)·H_i\left(f\right)·{\varphi }_i\left(x_P\right) .$$

(15)

The Fourier transform of the acceleration is calculated using the following formula:

$$\ddot{V}\left(x,f\right)=H_G\left(x,f\right)·\widehat{G}\left(f\right)+H_P\left(x,f\right)·\widehat{P}\left(f\right).$$

(16)

Finally, assuming by hypothesis that the cross-correlation between the functions $\widehat{G}\left(f\right)$ and $\widehat{P}\left(f\right)$ is negligible, if not completely zero, the RMS of the acceleration response along the beam in a generic abscissa $x$ is obtained using the following formula:

$${\ddot{v}}_{RMS}\left(x\right)=\sqrt{ {\int }_0^{\infty }\left\{ {‖ H_G\left(x,f\right) ‖}^2·W_G\left(f\right)+{‖ H_P\left(x,f\right) ‖}^2·W_P\left(f\right) \right\}·df } ,$$

(17)

where $W_G\left(f\right)$ is the PSD of the function $G\left(t\right)$, which describes the time history of lateral distributed load $p\left(x,t\right)$, and $W_P\left(f\right)$ is the PSD of the concentrated force $P\left(t\right)$. The function $G\left(t\right)$ could represent, for example, the temporal variation of lift on an aeroplane wing due to atmospheric turbulence (which can induce upwash or downwash on a wing); in this case, $W_G\left(f\right)$ would be linked to the spectral density obtained by filtering and manipulating the experimentally acquired data during flights along a specific route, possibly considering the effects of flight altitude.

2.2.2. Uniform Cantilever Beam Subjected to Base Excitation

For the base-driven system, if ${\ddot{V}}_B\left(f\right)$ indicates the Fourier transform of ${\ddot{v}}_B\left(t\right)$ (for the definition of $f_{gi}\left(t\right)$, see Equation (A22)), the following applies:

$$F_{gi}(f)=-m·{\int }_0^L{\varphi }_i\left(x\right)·dx·{\ddot{V}}_B\left(f\right) ,$$

(18)

the transfer function of the total acceleration response is obtained as shown below:

$$\begin{array}{cc}& H_{Tot}\left(x,f\right)=1+{\displaystyle \sum _{i=1}^N}{\varphi }_i\left(x\right)·H_i\left(f\right)·\left(-{\overline{m}}_i \right) ,\\ & {\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \overline{m}}_i=m·{\int }_0^L{\varphi }_i\left(x\right)·dx .\end{array}$$

(19)

The Fourier transform of the total acceleration is calculated as follows:

$${\ddot{V}}_{Tot}\left(x,f\right)=H_{Tot}\left(x,f\right)·{\ddot{V}}_B\left(f\right) ,$$

(20)

and, finally, the RMS of the total acceleration response along the beam in a generic abscissa $x$ is calculated as follows:

$${\ddot{v}}_{Tot-RMS}\left(x\right)=\sqrt{ {\int }_0^{\infty }\left\{{‖ H_{Tot}\left(x,f\right) ‖}^2·W_{{\ddot{v}}_B}\left(f\right)\right\}·df },$$

(21)

where $W_{{\ddot{v}}_B}\left(f\right)$ is the PSD of the input acceleration at the root of the beam (the base).

2.3. Calculation of Engineering Quantities

Using Equations (8), (10), (11) and (14) and superposing the contributions of the vibration modes considered in the analysis, within the scope of validity of the adopted structural theory (in the present case, the Euler–Bernoulli beam model [7]), it is possible to statistically calculate all the technical quantities of interest; above all, in the case of a beam, the bending moment and the bending stresses.

2.3.1. Example (a): Uniform Cantilever Beam Subjected to a Concentrated Random Force

Considering as an example the case of the uniform cantilever beam loaded only with a concentrated force,

$$Q_i(f)=-{\displaystyle \frac{{\ddot{Q}}_i\left(f\right)}{{\left(2·\pi ·f\right)}^2}}=-{\displaystyle \frac{H_i\left(f\right)·{\varphi }_i\left(x_P\right)·\widehat{P}\left(f\right) }{{\left(2·\pi ·f\right)}^2}},$$

(22)

thus, the Fourier transform of the bending moment is given as follows:

$$M_a\left(x,f\right)=-E·I·\sum _{i=1}^N{Q}_i\left(f\right)·{\displaystyle \frac{d^2{\varphi }_i\left(x\right)}{d^{2}x}}={\displaystyle \frac{E·I}{{\left(2·\pi ·f\right)}^2}}·\left( \sum _{i=1}^N{H}_i\left(f\right)·{\varphi }_i\left(x_P\right)·{\displaystyle \frac{d^2{\varphi }_i\left(x\right)}{d^{2}x}} \right)·\widehat{P}\left(f\right) .$$

(23)

The RMS of the instantaneous response of the bending moment in a generic section along the beam is obtained as follows:

$$M_{a-RMS}\left(x\right)={\displaystyle \frac{E·I}{{\left(2·\pi \right)}^2}}·\sqrt{ {\int }_0^{\infty }\left\{{‖ {\displaystyle \frac{1}{f^2}}·\left(\sum _{i=1}^N H_i\left(f\right)·{\varphi }_i\left(x_P\right)·{\displaystyle \frac{d^2{\varphi }_i\left(x\right)}{d^{2}x}}\right) ‖}^2·W_P\left(f\right)\right\}·df } .$$

(24)

The number of modes, $N$, must be sufficiently high to reduce errors in the bending moment computation. The RMS of maximum bending stress can be calculated using Navier’s formula ($h/2$ is the distance of the fiber from the neutral axis of the beam section):

$${\sigma }_{a-RMS}\left(x\right)={\displaystyle \frac{E·h/2}{{\left(2·\pi \right)}^2}}·\sqrt{{\int }_0^{\infty }\left\{{‖ {\displaystyle \frac{1}{f^2}}·\left(\sum _{i=1}^N H_i\left(f\right)·{\varphi }_i\left(x_P\right)·{\displaystyle \frac{d^2{\varphi }_i\left(x\right)}{d^{2}x}}\right) ‖}^2·W_P\left(f\right)\right\}·df } .$$

(25)

2.3.2. Example (b): Uniform Cantilever Beam Subjected to Random Base Excitation

In the base-driven problem,

$$Q_i\left(f\right)=-{\displaystyle \frac{{\ddot{Q}}_i\left(f\right)}{{\left(2·\pi ·f\right)}^2}}=-{\displaystyle \frac{H_i\left(f\right)·\left(-{\overline{m}}_i \right)·{\ddot{V}}_B\left(f\right) }{{\left(2·\pi ·f\right)}^2}} ,$$

(26)

now, the Fourier transform of the bending moment is given as follows:

$$M_b\left(x,f\right)=-E·I·\sum _{i=1}^N{Q}_i\left(f\right)·{\displaystyle \frac{d^2{\varphi }_i\left(x\right)}{d^{2}x}}=-{\displaystyle \frac{EI}{{\left(2·\pi ·f\right)}^2}}·\left( \sum _{i=1}^N{H}_i\left(f\right)·{\overline{m}}_i·{\displaystyle \frac{d^2{\varphi }_i\left(x\right)}{d^{2}x}} \right)·{\ddot{V}}_B\left(f\right) .$$

(27)

The RMS of the instantaneous response of the bending moment in a generic section along the beam is given as follows:

$$M_{b-RMS}\left(x\right)={\displaystyle \frac{E·I}{{\left(2·\pi \right)}^2}}·\sqrt{ {\int }_0^{\infty }\left\{{‖ {\displaystyle \frac{1}{f^2}}·\left(\sum _{i=1}^N H_i\left(f\right)·{\overline{m}}_i·{\displaystyle \frac{d^2{\varphi }_i\left(x\right)}{d^{2}x}}\right) ‖}^2·W_{{\ddot{v}}_B}\left(f\right)\right\}·df } .$$

(28)

The RMS of the maximum bending stress is now given as follows:

$${\sigma }_{b-RMS}\left(x\right)={\displaystyle \frac{E·h/2}{{\left(2·\pi \right)}^2}}·\sqrt{ {\int }_0^{\infty }\left\{{‖ {\displaystyle \frac{1}{f^2}}·\left(\sum _{i=1}^N H_i\left(f\right)·{\overline{m}}_i·{\displaystyle \frac{d^2{\varphi }_i\left(x\right)}{d^{2}x}}\right) ‖}^2·W_{{\ddot{v}}_B}\left(f\right)\right\}·df } .$$

(29)

3. Results

The methodology summarized in the previous paragraph is used to solve, by way of example, two cases which, although elementary from an engineering point of view, nevertheless highlight the simplicity and clarity of the method and the role played by the structural model and the descriptive model of non-deterministic loads.

It is evident from the mathematical formulation presented above that the elements necessary for the execution of the actual calculations concern, in the case of a beam, the following:

• The geometric data and mechanical characteristics of the beam;
• Knowledge of the first natural vibration frequencies of the beam;
• Knowledge, possibly explicit, of the first natural modes of vibration of the beam;
• The type of loading condition and, consequently, the knowledge of the PSD of the loading parameter.

Examples (a) and (b), preliminarily presented in the previous paragraph, are examined below.

The geometric data and mechanical characteristics of the beam are summarized in

Table 1. Geometric data and mechanical characteristics of the examined beam.

$\mathit{L}$ [m]	$\mathit{b}$ 1 [m]	$\mathit{h}$ 1 [m]	$\mathit{t}$ 1 [m]	$\mathit{I}$ 1 [m4]	$\mathit{E}$ [Pa]	$\mathit{m}$ [kg/m]
4.0	0.2	0.4	0.005	1.274 × 10−4	2.06 × 1011	46.02

¹ See Figure 1b.

(the material is a construction steel alloy).

The calculation of the natural frequencies and the related vibration modes of the uniform cantilever beam was performed using solutions available in the literature [55] (pp. 359–361).

Appendix C summarizes the formulas used: the second derivative of the vibration modes on which the calculation of the bending moment along the beam depends is also reported in full.

The extracted results concern the spectral density of displacement, acceleration, bending moment, and bending stress (for a classical slender beam model without effects of shear). These quantities can be calculated in a generic section along the beam span. For brevity, the formulas for the spectral density of displacement and acceleration have not been explicitly reported in the previous paragraph. However, they can be easily obtained from the expressions of the transfer functions and from the knowledge of the spectral densities of the load conditions considered (see Appendix D).

All calculations were performed very simply by preparing two Excel^® worksheets, available as Supplementary Materials; however, due to truncation errors in calculating the formulas for the higher order modes, only the results of the first 11 natural vibration modes were considered valid. It will be seen, however, that the calculations, although limited to the first 11 modes of vibration, are more than sufficient to guarantee a correct solution for the two proposed examples.

Table 2. Natural frequencies and coefficients of the uniform cantilever beam.

i	1	2	3	4	5	6	7	8	9	10	11
${\beta }_i$ [1/m]	0.4688	1.1735	1.9638	2.7489	3.5343	4.3197	5.1051	5.8905	6.6759	7.4613	8.2467
${\alpha }_i$	0.7341	1.0185	0.9992	1.0	1.0	1.0	1.0	1.0	1.0	1.0	1.0
$f_{ni}$ [Hz]	26.42	165.52	463.57	908.4	1501.6	2243.1	3133.0	4171.1	5357.6	6692.3	8175.4

shows the natural frequencies $f_{ni}$ and the ${\beta }_i$ and ${\alpha }_i$ coefficients (rounded values) calculated using the formulas in Appendix C.

3.1. Results of Example (a)

Let us now consider the case represented in Figure 2, assuming, to simplify the example, that the component of the distributed load, $r\left(x\right)$, is identically zero. The only active load component is, therefore, the force $P\left(t\right)$ applied at $x_P=L$.

Assuming, as previously stated, that we are dealing with stationary and ergodic random processes, the non-deterministic loading condition can be represented by a spectral density, shown, as an example, in Figure 4.

Using the methodology described in the previous paragraph, it is possible to calculate the dynamic response of the cantilever beam, a response that will also be stationary and ergodic.

Although in spreadsheets it is possible to assign different values to the damping factor as the frequency varies, if known, the calculations are performed with a constant value of ${\zeta }_i=\zeta =0.01$. This value agrees with the assumptions made in Section 2.2: assumptions valid for lightly damped systems, as indicated in [1].

The assumed spectral density (auto-PSD) of the load $P\left(t\right)$ (Figure 4) varies for frequencies between 10 Hz and 2000 Hz (this frequency range is typical of random vibration environments).

The dynamic results depend on only the first five modes of vibration of the cantilever (see Table 2).

The acceleration and displacement results (Figure 5 and Figure 6) are calculated at the free end of the beam $\left(x=L\right)$, while the bending moment and maximum bending stress results (Figure 7 and Figure 8) refer to the root section of the beam $\left(x=0\right)$.

For convenience,

Table 3. Example (a): RMS of load parameter and output parameters.

${\mathit{P}}_{\mathit{R}\mathit{M}\mathit{S}}$ [N]	${\mathit{v}}_{\mathit{a}-\mathit{R}\mathit{M}\mathit{S}}\left(\mathit{L}\right)$ [mm]	${\ddot{\mathit{v}}}_{\mathit{a}-\mathit{R}\mathit{M}\mathit{S}}\left(\mathit{L}\right)$ [m/s2]	${\mathit{M}}_{\mathit{a}-\mathit{R}\mathit{M}\mathit{S}}\left(0\right)$ [Nm]	${\mathit{\sigma }}_{\mathit{a}-\mathit{R}\mathit{M}\mathit{S}}\left(0\right)$ [MPa]
1006.56	1.13	229.89	7219.59	11.33

shows the calculated data for the effective value (RMS) of the applied load and the calculated quantities. All RMS values were calculated by taking the square root of the area under the spectral density curves. The trapezoidal rule was used to calculate the integrals; for this reason, the data must be considered as estimated.

It must be kept in mind that the values of the bending moment and the maximum bending stress consider the inertial effects due to the oscillation of the beam. Even if the RMS of the applied force is not particularly high for the examined beam (about 1006 N), the RMS of acceleration of the beam end section reaches about 223 m/s² (about 23.4 g; g is the acceleration of gravity at sea level).

3.2. Results of Example (b)

The second example concerns the loading condition of Figure 3. The excitation is given by the acceleration of the base ${\ddot{v}}_B\left(t\right)$. The acceleration of the base, a stationary and ergodic random process, is assigned via the spectral density (auto-PSD) shown in Figure 9.

The frequencies vary between 10 Hz and 2000 Hz, so the results depend only on the first five natural modes of vibration. The damping factor is constant: $\zeta =0.01$.

The results of both relative and total acceleration and displacement (Figure 10, Figure 11, Figure 12 and Figure 13) are calculated at the free end of the beam $\left(x=L\right)$.

The bending moment and maximum bending stress results (Figure 14 and Figure 15) refer to the root section of the beam $\left(x=0\right)$.

Table 4. Example (b): RMS of base acceleration and main output parameters.

${\ddot{\mathit{v}}}_{\mathit{B}-\mathit{R}\mathit{M}\mathit{S}}$ [m/s2]	${\mathit{v}}_{\mathit{b}-\mathit{T}\mathit{o}\mathit{t}-\mathit{R}\mathit{M}\mathit{S}}\left(\mathit{L}\right)$ [mm]	${\ddot{\mathit{v}}}_{\mathit{b}-\mathit{T}\mathit{o}\mathit{t}-\mathit{R}\mathit{M}\mathit{S}}\left(\mathit{L}\right)$ [m/s2]	${\mathit{M}}_{\mathit{b}-\mathit{R}\mathit{M}\mathit{S}}\left(0\right)$ [Nm]	${\mathit{\sigma }}_{\mathit{b}-\mathit{R}\mathit{M}\mathit{S}}\left(0\right)$ [MPa]
51.55	3.61	266.60	21,228.37	33.31

shows the (approximately) calculated data of the effective (RMS) value of the base acceleration and some of the other output quantities.

In this example, the effect of the base acceleration is evident: the RMS of the absolute acceleration of the beam end section reaches 266.60 m/s² (about 27 g).

The RMS of the maximum bending stress this time is not negligible (about 33 MPa), much higher than the previous example.

It is worth remembering that the effective value (for random vibrations with zero mean value) represents a quantity having, for a Gaussian distribution, a “statistical significance” of approximately 68.3% (which corresponds to the probability that the instantaneous response is, in absolute value, less than or equal to the calculated RMS); therefore, to increase the reliability of the obtained results, this value is normally multiplied, for example, by two, three, or more, depending on whether one wants to adopt the 2σ criterion (about 95.5%), the 3σ criterion (about 99.7%), or the approach based on the distribution of extreme value [33] (pp. 359–362).

Whatever the criterion adopted, σ is the standard deviation of the Gaussian instantaneous response.

4. Discussion

The presented methodology allows us to examine, in a not particularly complex way, technical problems of structural engineering that concern beam-like models subjected to non-deterministic load conditions.

An important step in the analysis consists in the calculation of the modal characteristics of the structure (frequencies and natural modes of vibration). For straight, slender, uniform beams and in the presence of not particularly complex constraint conditions, closed-form solutions can be used. In this case, calculation of the generalized coefficients of the equations of motion is quite easy: this is what has been demonstrated in this work.

Then, starting from the knowledge of the spectral density of the load parameter, be it a tip force or an acceleration applied to the base section of a cantilever beam, the dynamic responses were explicitly calculated in terms of displacement, acceleration, bending moment and maximum bending stress in a generic section of the beam.

The application examples discussed clearly show the effect of the load data and clearly show whether and how the beam vibration modes participate in the dynamic response.

The calculation procedure illustrated is intended to highlight the flexibility and clarity of the method. With much more structured and effective calculation codes, it is certainly possible to conduct parametric studies and address any optimization issues.

However, with the data generated using a simple spreadsheet, as is the case in this work, it would be possible, for example, to (1) set up a calculation of fatigue damage of the beam material, (2) estimate whether the vibrations induced on the beam can be used in energy-harvesting systems, (3) calculate the acceleration levels to which instruments or electrical systems installed on the beam could be subjected, (4) define the effects of very high unexpected load peaks, (5) study in first approximation the effects of a gust on the wing structure of an aircraft, and (6) study in first approximation the effects of earthquakes or wind profiles on the structure of a skyscraper.

The methodology presented does not consider the phenomena of spatial cross-correlation of the load data as, in the cases examined, the dynamic response can be studied starting from the sole knowledge of the input spectral density by calculating the autocorrelated response parameters.

The case of space- and time-varying loads (as happens, for example, in bridge structures, in the case of aerodynamic loads due to highly turbulent flows, or in the case of aeroacoustic pressure fields) requires a slightly more complicated formulation; above all, it requires complete knowledge of the spectra that translates into the knowledge of as many functions or data tables that describe both auto-PSDs and cross-PSDs [35].

The literature contains many examples of solutions for beams subjected to non-deterministic loads, especially in the bridge construction sector. In these cases, however, the mobile loads on the structures require, as said above, the use of more laborious and less flexible analytical techniques (see, for example, [8,17]).

In the case of non-uniform beams or beams for which it is necessary to consider the shear deformability effects, it is possible to apply solution methods based on the assumed modes technique [2] (pp. 282–290). In this case, as is well known, it is necessary to pay close attention to the choice of the type and number of functions assumed, correctly considering the constraint conditions of the beam.

Regarding the examples discussed in this work, both numerical solutions based on the application of the finite element method and analytical solutions are available in the literature [6,7], which often refer to more complex cases [56,57,58,59] but, nevertheless, do not allow for a simple and direct verification of the effects of the design parameters and almost never propose comparisons with experimental measurements but only comparisons with other complex numerical approaches.

In the case of small structural damping, the approach followed in the present work, which is based on the use of explicit mathematical relations linking the various physical parameters that define the problem of the dynamic response of beam-like structures, allows us to develop, in its simplicity, a technically useful tool (especially in the preliminary design phase, when the configuration of the structures is not completely defined and the analysis times must be sufficiently rapid) that is precise and, at the same time, possibly economical.

Possible future developments of this activity will concern the application of the methodology to models of wing structures or blades of wind turbines and the preliminary implementation of the method with structural elements that can be schematized as assemblies of plates or sandwich-like structures, since the characteristic of being structurally lightweight remains one of the technical aspects of greatest interest in modern engineering [60].

A further aspect that could be developed concerns the possibility of considering the nonlinearities due to the constitutive characteristics of the material (rate-independent hysteresis effects linked to possible repeated high levels of deformation). For this purpose, the authors of [61,62] developed an analytical methodology for the closed-form representation of the rate-independent but strongly nonlinear behavior of the material and for the time domain solution of the dynamic equations of mechanical systems affected by such phenomena. The extension of the method proposed in [61,62] to systems with multiple degrees of freedom and especially the definition of dynamic responses in the frequency domain would lead to the generalization of the method presented in the present work, which is currently applicable to the case of linear structural systems.

5. Conclusions

• In this work, based on the following classical hypotheses
• linear elastic, uniform, straight, and slender beam,
• homogeneous and isotropic material,
• stationary and ergodic random processes,

and assuming that the system is lightly damped, engineering parameters of the random response of a cantilever beam model subjected to random loads were obtained in explicit form.

Two conditions of random loading have been studied in detail:

• (a) Cantilever beam loaded with a lateral force applied at its free end;
• (b) Cantilever beam subjected to forced vibrations of the root section.
• The analytical expressions of the transfer functions for acceleration, displacement, bending moment, and bending stress were calculated as a function of external lateral loads. The loading conditions were assigned in terms of power spectral densities.

The PSD of the input loads can be defined in the design specifications referring to well-defined operating environments, or it can be obtained by processing the data available in the technical user manuals, as in the case of satellite launchers, which are made available by the manufacturers of the launchers themselves (see Sections 3.2.5 and 3.2.6 of [63]).

The results clearly show the effects of the physical parameters on which the random dynamic response of the beam depends, highlighting, above all, the role of the natural vibration frequencies of the beam.

It should be noted that the effects of g, the acceleration of gravity, were not taken into consideration.

The results obtained, available in the form of Supplementary Materials, provide the spectral density of the responses (displacement, acceleration, bending moments, and maximum bending stress) both in graphical and tabular form in a generic section of the beam. In the calculation sheets, it is possible to introduce different values of the damping factor for each natural mode of vibration.

By integrating (in the frequency domain) the spectral density of the individual response parameters, the RMS of these parameters is obtained, which as is known has an explicit technical value and a precise probabilistic connotation for purely random processes (approximately 68%).

The calculations discussed in this paper were performed using worksheets. These worksheets, within the hypotheses of validity of the method, allow us to perform preliminary parametric analyses by varying the input data (geometric data, material characteristics, and PSD of the loads).