## Abstract

Beam structures are widely used in various engineering occasions to model various structures. Numerous researchers have studied dynamic responses of beam structures with nonlinear supports or nonlinear foundations. In engineering, nonlinear supports were subjected to the beam structure through the surface contact rather than the point connection. Few works studied the dynamic behavior of the beam structure with local uniform cubic nonlinear stiffness foundations. Additionally, the boundary rotational restraints of the beam structure are ignored. To improve the engineering acceptance of the beam structure with nonlinearity, it is of great significance to study the dynamic behavior of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation. This work establishes a nonlinear dynamic model of the beam structure with a local uniform nonlinear foundation. Dynamic responses of the beam structure are predicted through the Galerkin truncated method. In Galerkin truncated method, mode functions of the axially loaded beam structure without the local uniform nonlinear foundation are selected as the trail and weight functions. The harmonic balance method is employed to verify the correctness of the Galerkin truncated method. The influence of the sweeping ways and local uniform nonlinear foundation on dynamic responses of the generally restrained axially loaded beam structure is investigated. Dynamic responses of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation are sensitive to its calculation initial values. Suitable parameters of the local uniform nonlinear foundation can suppress the vibration response and transform the vibration state of the beam structure.

## 1 Introduction

On various engineering occasions, beam structures are widely employed to model complex structures according to their excellent structural characteristics. External excitations introduced by the power equipment bring the unwanted vibration of beam structures, then cause structural destruction, equipment damage, and other serious problems. A deep understanding of the vibration characteristics of beam structures is the premise to suppress the unwanted vibration of relevant complex structures.

Numerous research efforts have been devoted to the vibration characteristics of beam structures under different boundary conditions. Kang and Kim [1] summarized the previous studies related to beam structures and studied the modal properties of the beam structure with general boundary conditions, where rotational and translational boundary springs were introduced to simulate the boundary condition. For improving the engineering practicability of beam structures, the Fourier series was applied to predict the modal properties of beam structures under classical boundary conditions [2,3]. The traditional Fourier series was discontinuous at the boundary in predicting the modal properties of beam structures, which limited their applications in engineering practice. To improve the continuity at the boundary of the traditional Fourier series, Li [4] introduced four auxiliary terms into the traditional Fourier series. The improved Fourier series was smooth in the entire domain including the boundaries, which was also regarded as the boundary-smoothed Fourier series. By using the boundary-smoothed Fourier series, numerous researchers studied vibration characteristics of various beam structures with general boundary conditions, including the laminated composite beam [5], the acoustic black hole beam [6], the rotating beam [7], the rotating beam with distributed piezoelectric modal sensor [8], the axially loaded beam [9], and the axially loaded beam with nonuniform supports [10]. The above research verified the correctness of the boundary-smoothed Fourier series in predicting vibration characteristics of elastic beam structures under general boundary conditions.

In the marine engineering branch, the propulsion shaft is simplified as a beam structure in analyzing its vibration characteristics and dynamic responses. In engineering practice, the propulsion shaft is typically subjected to axial loads introduced by the power equipment and propellor. The pressure load harms the stability of the propulsion shaft. Furthermore, for the shafting systems employed in marine engineering, the axial motion of the shaft is also harmful to the stability of the shafting systems. Therefore, some limiters are installed on the shafting systems to limit their axial motion. Such limiters can be simplified as the axial load which is subjected to the shafting system. To improve the structural stiffness of the vibration system, internal supporting bearings are installed into the propulsion shaft. Additionally, isolators, absorbers, and other equipment are artificially introduced to suppress the vibration level of the propulsion shaft. The internal bearings and isolators are simplified as the translational stiffness and concentrated mass. For the beam structure with internal supports, vibration characteristics of the beam structure with intermediate supports were investigated in Refs. [11,12]. Darabi et al. [13] investigated the vibration characteristics of the beam–mass–spring system. Burlon et al. [14] performed the exact frequency response analysis in predicting the dynamic responses of the axially loaded beam with viscoelastic dampers. However, the above research is mainly limited to linear cases.

In engineering practice, the equivalent stiffness of the internal supporting bearing is significantly influenced by its oil film thickness, fit clearance, installation mode, and others. The equivalent stiffness of the internal supporting bearing is nonlinear when the working condition of the internal supporting bearing changes. Additionally, some nonlinear mechanisms [1518] are designed to suppress structural vibration. Such nonlinear mechanisms can be introduced into the propulsion shaft artificially.

For the beam structure with nonlinear supports, Pakdemirli and Boyaci [19] investigated the nonlinear vibration responses of a beam structure with nonideal supports, in which the boundary condition was simply–simply supported. Ghayesh et al. [2023] deeply investigated the nonlinear vibration and potential engineering applications of various beam structures with cubic nonlinearities, internal boundary conditions, and nonlinear springs. Wang and Fang [24] studied amplitude–frequency response curves of the beam structure with nonlinear supports at both ends, where cubic springs were employed to simulate the nonlinear boundary conditions. Mao et al. [25] established a nonlinear vibration analysis model of the flexible beam structure under nonlinear boundary conditions. The dynamic behavior of such a vibration analysis model was predicted through the method of multiple scales. Ding et al. [26] established a nonlinear vibration analysis model of the pre-pressure beam structure supported by a type of adjustable nonlinear isolators and firstly studied the influence of the quasi-zero-stiffness system on the isolation of multimodal vibration of elastic structures. Ghayesh [2729] systemically established the nonlinear vibration analysis model for the axially functionally graded and multilayered beams and studied their nonlinear dynamic responses. Ding and Chen [30] established a nonlinear vibration analysis model of a slightly curved beam with quasi-zero-stiffness isolators and investigated the isolation of the bending vibration of the initial curved structure. Burlon et al. [31] employed statistical linearization with constrained modes to investigate nonlinear random vibrations of beams with multiple internal supports. With the consideration of engineering practice, Zhao and Du [32] established the nonlinear vibration analysis model of an axially loaded beam structure supported by a spring–mass system. The influence of the boundary and internal nonlinearity on the dynamic behavior of beam structure was investigated. In the above studies, the nonlinear supports were mainly subjected to the beam structure in the pattern of point connection. However, the nonlinear supports were subjected to the beam structure through the surface contact in engineering practice. To study the dynamic behavior of beam structures with nonlinear supports more accurately, it is of great significance to establish the analysis model of beam structures with nonlinear foundations.

For the beam structure with nonlinear foundations, Tsiatas [33] studied the nonlinear vibration of the nonuniform beam structure under nonlinear elastic foundations through the AEM solution. Ma et al. [34] investigated the nonlinear free vibration characteristics of the beam structure on elastic foundations. The nonlinear vibration characteristics were obtained by using the eigenvalue analysis method and the method of multiple scales. Essa [35] predicted the dynamic behavior model of elastic beams with linear and nonlinear foundations by using the finite difference method, where the boundary conditions of the beam structure were classical. Dang and Le [36] investigated the nonlinear vibration of the Euler–Bernoulli beam structure with a Winkler foundation through the equivalent linearization method. The elastic nonlinear foundations in the above research were typically modeled as the Winkler foundation. Few works studied the dynamic behavior of the beam structure with local uniform cubic nonlinear stiffness foundations. Additionally, boundary rotational restraints of the beam structure were ignored in most studies, limiting their engineering application.

Considering the engineering practice, this work studies the nonlinear vibration of a generally restrained axially loaded beam structure with a local uniform nonlinear foundation, where the axial motion of the beam structure is ignored and the effects of the nonlinear foundation only considered in the transverse direction. The dynamic behavior of the beam structure is predicted through the Galerkin truncated method. The harmonic balance method is employed to verify the correctness of the Galerkin truncated method. On this basis, the influence of the sweeping ways of the external excitation and parameters of the local uniform nonlinear foundation on the dynamic behavior of the generally restrained axially loaded beam structure is investigated. Finally, some conclusions are drawn.

## 2 Theoretical Formulations

This section studies the theoretical formulations of a generally restrained axially loaded beam structure with a local uniform nonlinear foundation. The nonlinear vibration analysis model of the beam structure with a local uniform nonlinear foundation is established. The Galerkin truncated method is utilized to predict the dynamic behavior of the beam structure with a local uniform nonlinear foundation.

### 2.1 Model Description.

As illustrated in Fig. 1, the physical model of a generally restrained axially loaded beam structure with a local uniform nonlinear foundation is established. The vibration system in Fig. 1 consists of a beam structure, boundary restraints, and a local uniform nonlinear foundation. Considering the engineering practice, ship propulsion shafting systems typically consist of multiple slender shafts. When studying the transverse vibration of such a shafting system, the slender shafting system can be simplified as the Euler–Bernoulli beam. Therefore, the Euler–Bernoulli beam is applied to model the beam structure in this study. The transverse vibration displacement of the beam structure is u(x, t). E, CB, ρ, S, L, and I are modulus of elasticity, viscous damping, mass density, section area, length, and inertia moment of the beam structure, respectively. P is the axial load subjected to the beam structure. Boundary translational and rotational restraining springs are introduced into the beam structure to simulate the elastic boundary conditions. Various boundary conditions can be simulated by setting the stiffness coefficient of boundary springs. kL and kR are the stiffness coefficients of the boundary translational restraining springs. KL and KR are the stiffness coefficients of the boundary rotational restraining springs. The local uniform nonlinear foundation consists of distributed mass mI, viscous damping CI, linear stiffness kI, and nonlinear stiffness knI. The center position of the uniform nonlinear foundation is xI and Lf is the length of the uniform nonlinear foundation.

Fig. 1
Fig. 1
Close modal
According to Newton’s second law and vibration theory, the nonlinear governing equation of the beam structure with a local uniform nonlinear foundation is written as
$EI∂4u∂x4+P∂2u∂x2+ρS∂2u∂t2+CB∂u∂t+δ(x−xF)F0sin(ωt)+G(x)(mI∂2u∂t2+CI∂u∂t+kIu+knIu3)=0$
(1)
where G(x) is defined as
$G(x)={0,0≤x
(2)
The restoring force introduced by the local uniform nonlinear foundation is derived as
$FN=∫xI−Lf/2xI+Lf/2(CI∂u∂t+kIu+knIu3)dx$
(3)
In this study, there is a harmonic point excitation located at the beam structure. F0 is the amplitude of the harmonic point excitation. ω is the angular frequency of harmonic point excitation. xF is the position of harmonic point excitation. The specific form of the harmonic point excitation is written as
$F(x,t)=δ(x−xF)F0sin(ωt)$
(4)
Considering the engineering practice, the effect of the axial load on the axial displacement on the boundary condition is ignored. The boundary conditions at both ends of the beam structure are derived as
$x=0:{kLu+EI∂3u∂x3+P∂u∂x=0KL∂u∂x−EI∂2u∂x2=0$
(5a)
and
$x=L:{kRu−EI∂3u∂x3−P∂u∂x=0KR∂u∂x+EI∂2u∂x2=0$
(5b)

Then, the Galerkin truncated method can be utilized to discretize and solve the nonlinear governing equation of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation.

### 2.2 Solution Procedure.

In this section, the Galerkin truncated method is applied to predict the nonlinear dynamic behavior of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation. Mode functions of a generally restrained axially loaded beam structure without the local uniform nonlinear foundation are a set of functions that satisfy the boundary conditions of the beam structure analyzed in Sec. 2.1. In the Galerkin truncated method, such mode functions are chosen as the trial and weight functions. The Galerkin condition is utilized to discretize the nonlinear governing equation of the beam structure. Then, the residual equations of the beam structure are established and solved through the numerical method. The nonlinear dynamic behavior of the beam structure can be obtained by arranging the calculation results of the residual equations.

Before applying the Galerkin discretization, the transverse vibration displacement of beam structure with a local uniform nonlinear foundation is expanded as the form of mode superposition, namely,
$u(x,t)=∑i=1Nφi(x)qi(t)$
(6)
where N is the truncated number, φi(x) is the ith trial function, and qi(t) is the ith term related to time. In this study, mode functions of a generally restrained axially loaded beam structure are chosen as the trail functions. Correspondingly, φi(x) is also the ith mode function of the generally restrained axially loaded beam structure. Typically, such mode functions can be effectively gained by applying the boundary-smoothed Fourier series with the energy principle [9].
Applying the Galerkin condition to Eq. (3), the mth residual equation of the beam structure is derived as
$∫0L[EI∂4u∂x4+P∂2u∂x2+ρS∂2u∂t2+CB∂u∂t+δ(x−xF)F0sin(ωt)+G(x)(mI∂2u∂t2+CI∂u∂t+kIu+knIu3)]ψm(x)dx$
(7)
where ψm(x) is the mth weight function and m = 1,2, … M. In this study, mode functions of a generally restrained axially loaded beam structure are chosen as the weight functions. ψm(x) is also the mth mode function of the generally restrained axially loaded beam structure.
Then, substituting Eq. (6) into Eq. (7) and simplifying the mth residual equation, the rearranged mth residual equation of the beam structure is derived as
$R1m+R2m+R3m+R4m+R5m+R6m+R7m+R8m+R9m=0$
(8)
The specific form of each term in Eq. (8) is as follows:
$R1m=∫0L[EI∑i=1Nd4φi(x)dx4qi(t)]ψm(x)dx$
(9a)
$R2m=∫0LP[∑i=1Nd2φi(x)dx2qi(t)]ψm(x)dx$
(9b)
$R3m=∫0LρS[∑i=1Nφi(x)d2qi(t)dt2]ψm(x)dx$
(9c)
$R4m=∫0LCB[∑i=1Nφi(x)dqi(t)dt]ψm(x)dx$
(9d)
$R5m=F0sin(ωt)ψm(xF)$
(9e)
$R6m=∫xI−Lf/2xI+Lf/2mI[∑i=1Nφi(x)d2qi(t)dt2]ψm(x)dx$
(9f)
$R7m=∫xI−Lf/2xI+Lf/2CI[∑i=1Nφi(x)dqi(t)dt]ψm(x)dx$
(9g)
$R8m=∫xI−Lf/2xI+Lf/2kI[∑i=1Nφi(x)qi(t)]ψm(x)dx$
(9h)
$R8m=∫xI−Lf/2xI+Lf/2knI[∑i=1Nφi(x)qi(t)]3ψm(x)dx$
(9i)
According to Eqs. (9d) and (9f), R4m and R6m are the terms related to the second derivate of the time. Putting R4m and R6m into another side of Eq. (8), Eq. (8) is changed as
$R4m+R6m=−(R1m+R2m+R3m+R5m+R7m+R8m+R9m)$
(10)
Expanding the expression of R4m and R6m, the specific form of R4m + R6m is written as
$R4m+R6m=B1md2q1dt2+⋯+Bimd2qidt2+⋯+BNmd2qNdt2$
(11)
where Bim is the constant coefficient.
To solve the residual equations of the beam structure effectively, it is of great significance to arrange the residual equations into the form of a matrix. Putting Eq. (11) into Eq. (10) and arranging the residual equations of the beam structure, the matrix form of the residual equations is derived as
$[B11…Bi1…BN1……………B1m…Bim…BNm……………B1M…BiM…BNM][d2q1dt2…d2qidt2…d2qNdt2]=−[RT1…RTm…RTM]$
(12)
in which RTm = R1m + R2m + R3m + R5m + R7m + R8m + R9m.
Since the coefficient matrix of the second derivate of the time in Eq. (12) is not a diagonal matrix, Eq. (12) cannot be directly solved through the numerical method. Therefore, applying the matrix transformation to Eq. (12), which is rewritten as
$[d2q1dt2…d2qidt2…d2qNdt2]=−[D11…D1m…D1M……………Di1…Dim…DiM……………DN1…DNm…DNM][RT1…RTm…RTM]$
(13)

Equation (13) can be solved directly through the numerical method. In this study, the Runge–Kutta method is utilized to solve Eq. (13). Substituting the calculation results of Eq. (13) into Eq. (6), dynamic responses of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation can be obtained.

## 3 Numerical Results and Discussion

This section exploits the simulation platform to study the theoretical formulations analyzed in Sec. 2. The Galerkin truncated method is programmed in the simulation platform to discrete the nonlinear governing equation of beam structure. The numerical method is programmed to solve the residual equations. Then, dynamic responses of the beam structure predicted by the Galerkin truncated method are verified through the comparison with those predicted by the harmonic balance method.

In marine engineering, for the power equipment, before reaching the rated working conditions, its speed typically undergoes a process of increasing or decreasing, where the change in the speed is corresponding to the external excitation frequency. It is worth noting that when the speed changes, the vibration responses introduced by the power equipment will briefly maintain their state before the change, which means that the initial vibration responses after the speed change of the power equipment are the vibration response before the speed change. Furthermore, the rotation speed of power equipment typically increases from zero to its rated rotation speed. In the above process, the excitation frequency introduced by the power equipment is increasing from zero to its rated excitation frequency. Eventually, to ensure the lifetime of the power equipment, the power equipment typically works in its rated condition, which indicates that the excitation frequency introduced by the energy equipment is determined. Considering the engineering practice, the influence of the sweeping ways of the external excitation and parameters of the local uniform nonlinear foundation on amplitude–frequency responses and single-frequency responses of the beam structure is investigated.

### 3.1 Model Validation.

In this section, the stability and correctness of the Galerkin truncated method in predicting the dynamic behavior of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation are studied. Considering the engineering practice, the external excitation is typically applied at the boundary of the vibration structure. The position of harmonic point excitation is set as xF = 0 m. The amplitude of the harmonic point excitation is set as F0 = 10 N. In this study, the excitation frequency of the external excitation varies from 1 to 200 Hz. Parameters of the beam structure and boundary conditions are tabulated in Table 1. Parameters of the local uniform nonlinear foundation are listed in Table 2.

Table 1

Parameters of the beam structure and boundary conditions

Modulus of elasticityE (Pa)6.89 × 1010
Densityρ (kg/m3)2.8 × 103
LengthL (m)0.5
Cross-sectional areaS (m2)2 × 10−4
Moment of inertiaI (m4)1.677 × 10−9
Viscous dampingCB (N s/m)10
Boundary translational stiffnesskL/kR (N/m)5 × 104/5 × 102
Modulus of elasticityE (Pa)6.89 × 1010
Densityρ (kg/m3)2.8 × 103
LengthL (m)0.5
Cross-sectional areaS (m2)2 × 10−4
Moment of inertiaI (m4)1.677 × 10−9
Viscous dampingCB (N s/m)10
Boundary translational stiffnesskL/kR (N/m)5 × 104/5 × 102
Table 2

Parameters of the local uniform nonlinear foundation

Distributed massmI (kg/m)1.25
Center positionxI (m)0.25
Length of the foundationLf (m)0.04
Viscous damping of the foundationCI (N s/m2)250
Linear translational stiffnesskI (N/m2)1.25 × 104
Nonlinear translational stiffnessknI (N/m4)5 × 1011
Distributed massmI (kg/m)1.25
Center positionxI (m)0.25
Length of the foundationLf (m)0.04
Viscous damping of the foundationCI (N s/m2)250
Linear translational stiffnesskI (N/m2)1.25 × 104
Nonlinear translational stiffnessknI (N/m4)5 × 1011
In the Galerkin truncated method, the calculation time domain is selected as 0 ∼ 500 Te. Te is the period of the harmonic point excitation, which is defined as
$Te=2πω$
(14)

To ensure the transient dynamic responses of the beam structure with a local uniform nonlinear foundation fully die away, dynamic responses of the beam structure in the 401 ∼ 500Te are selected as the stable steady-state responses. Additionally, dynamic responses of the beam structure with a local uniform nonlinear foundation in 401 ∼ 500Te can also be regarded as the long-time permanent stable steady-state responses. To ensure the coefficient matrix of Eq. (13) is a square matrix, the truncated number of the Galerkin truncated method is set as N = M.

First, the stability of the Galerkin truncated method in predicting the dynamic behavior of the generally restrained beam structure with a local uniform nonlinear foundation is studied. The stable-steady amplitude–frequency responses at both ends of the beam structure are plotted in Fig. 2, in which the truncated number of the Galerkin truncated number is set as 2-term, 4-term, and 6-term. From Fig. 2, amplitude–frequency response curves stay stable as the truncated number reaches 4-term. In the subsequent study, the truncated number of the Galerkin truncated method is selected as N = M = 4. Additionally, a jumping down phenomenon appears in the stable steady-state amplitude–frequency response curves at both ends of the beam structure due to the existence of the local uniform nonlinear foundation.

Fig. 2
Fig. 2
Close modal
Second, the correctness of the Galerkin truncated method in predicting the dynamic response of the beam structure with a local uniform nonlinear foundation is investigated. The harmonic balance method is employed to verify the dynamic responses predicted by the Galerkin truncated method. Considering the form of the nonlinearity, the solutions of the harmonic balance method are set as
$q1=A1cos(ωt)+A2sin(ωt)+A3cos(3ωt)+A4sin(3ωt)$
(15a)
$q2=B1cos(ωt)+B2sin(ωt)+B3cos(3ωt)+B4sin(3ωt)$
(15b)
$q3=C1cos(ωt)+C2sin(ωt)+C3cos(3ωt)+C4sin(3ωt)$
(15c)
and
$q4=C1cos(ωt)+C2sin(ωt)+C3cos(3ωt)+C4sin(3ωt)$
(15d)
where Ai, Bi, Ci, and Di are the undetermined coefficients.

Stable steady-state amplitude–frequency response curves at both ends of the beam structure with a local uniform nonlinear foundation are plotted in Fig. 3. From Fig. 3, amplitude–frequency response curves at both ends of the beam structure predicted by the Galerkin truncated method and harmonic balance method match well with each other, which verifies the correctness of the Galerkin truncated method in predicting the dynamic behavior of the generally restrained beam structure with a local uniform nonlinear foundation.

Fig. 3
Fig. 3
Close modal

In summary, the Galerkin truncated method has good stability and correctness in predicting the dynamic behavior of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation.

### 3.2 Influence of the Excitation Sweeping Ways on Beam Dynamic Responses.

This section studies the influence of the sweeping ways of the external excitation on steady-state amplitude–frequency response curves at both ends of the generally restrained axially loaded beam with a local uniform nonlinear foundation. During the variation of excitation frequency, the end values of vibration response under the previous harmonic excitation frequency are employed as the initial values of the vibration response under the current harmonic excitation. Parameters of the beam structure, boundary conditions, and the external harmonic point excitation are the same as those in Sec. 3.1.

First, the influence of the sweeping ways on stable steady-state amplitude–frequency response curves at both ends of the beam structure under knI = 1.25 × 1012 N/m4 is studied, where xI = 0.3 m. Other parameters of the local uniform nonlinear foundation are the same as those in Table 2. Stable steady-state amplitude–frequency response curves at both ends of the beam structure with forward and reverse sweeping ways (knI = 1.25 × 1012 N/m4) are plotted in Fig. 4. From Fig. 4, a jumping down phenomenon can be observed in amplitude–frequency response curves at both ends of the beam structure. The jumping down of amplitude–frequency response curves appears at 115 Hz for the forward sweeping way. The jumping down of amplitude–frequency response curves appears at 112 Hz for the reverse sweeping way. For the left end, the peak of the second primary resonance region in the forward sweeping way is greater than that of the second primary resonance region in the reverse sweeping way. In amplitude–frequency response curves, several regions have multiple amplitudes. To further study the vibration state of such regions, phase diagrams of the targeted regions are plotted. Meanwhile, Poincaré points are also plotted in each phase diagram. When the harmonic excitation frequency is 10 Hz and 36 Hz, there is only one Poincaré point in each phase diagram and the phase path stays stable, which implies that the vibration state at 10 Hz and 36 Hz is in the single-periodic state.

Fig. 4
Fig. 4
Close modal

Second, the influence of the sweeping ways on stable steady-state amplitude–frequency response curves at both ends of beam structure under knI = 2 × 1012 N/m4 is studied. Other parameters of the local uniform nonlinear foundation are the same as those employed in Fig. 4. Stable steady-state amplitude–frequency response curves at both ends of the beam structure with forward and reverse sweeping ways (knI = 2 × 1012 N/m4) are plotted in Fig. 5. From Fig. 5, a jumping down phenomenon can be observed in amplitude–frequency response curves at both ends of the beam structure. The jumping down of the amplitude–frequency response curves appears at 120 Hz for the forward sweeping way. The jumping down of the amplitude–frequency response curves appears at 113 Hz for the reverse sweeping way. For the amplitude–frequency response curves at the left end of the beam structure, the peak of the second primary resonance region in the forward sweeping way is greater than that of the second primary resonance region in the reverse sweeping way. Additionally, in the amplitude–frequency response curves, several regions have multiple amplitudes. To further study the vibration state of such regions, phase diagrams of the targeted regions are plotted in Fig. 5. Meanwhile, Poincaré points are also plotted in each phase diagram. When the harmonic excitation frequency is 10 Hz and 36 Hz, there is only one Poincaré point in each phase diagram and the phase path stays stable. The above phenomenon implies that the vibration state at 10 Hz and 36 Hz is in the single-periodic state. When the harmonic excitation frequency is 115 Hz, the Poincaré points form a closed curve and the phase path stays stable. Therefore, the vibration state at 115 Hz is in the quasi-periodic state.

Fig. 5
Fig. 5
Close modal

Through the comparison of amplitude–frequency response curves in Figs. 4 and 5, it can be concluded that the increase of nonlinear stiffness of the local uniform nonlinear foundation enhances the difference between the amplitude–frequency response curves in the forward and reverse sweeping ways. Dynamic responses of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation are sensitive to the calculation of initial values. Additionally, the complex dynamic behavior of beam structure is motivated by certain nonlinear stiffness of the local uniform nonlinear foundation.

### 3.3 Influence of the Local Uniform Nonlinear Foundation on Amplitude–Frequency Responses.

This section studies the influence of the local uniform nonlinear foundation on the amplitude–frequency response curves at both ends of the beam structure, where the amplitude–frequency response curves are obtained through the forward sweeping way. Parameters of the initial values of the initial harmonic excitation are kept the same as those used in Sec. 3.2. Parameters of the beam structure and boundary conditions are given in Table 1.

First, the influence of nonlinear stiffness on the amplitude–frequency response curves at both ends of the beam structure is studied. Amplitude–frequency response curves at both ends of the beam structure under different nonlinear stiffness are plotted in Fig. 6, where the nonlinear stiffness is set as 0 N/m4, 2.5 × 1011 N/m4, 7.5 × 1011 N/m4, 1.25 × 1012 N/m4, 1.75 × 1012 N/m4, and 2.25 × 1012 N/m4, respectively. Other parameters of the local uniform nonlinear foundation are listed in Table 2. From Fig. 6, the first and second primary resonance regions shift to the higher frequency regions with the increase of the nonlinear stiffness. Meanwhile, the jumping down of the amplitude–frequency response curves appears in the above process. Compared with the peaks of amplitude–frequency response curves at both ends of the beam structure without nonlinear stiffness, the increase of the nonlinear stiffness has a beneficial effect on the vibration suppression at both ends of the beam structure.

Fig. 6
Fig. 6
Close modal

Second, the influence of the viscous damping of the local uniform nonlinear foundation on amplitude–frequency response curves at both ends of the beam structure is studied. Amplitude–frequency response curves at both ends of the beam structure under different viscous damping are plotted in Fig. 7, where the viscous damping of the local uniform nonlinear foundation is set as 25 N s/m2, 125 N s/m2, 250 N s/m2, 500 N s/m2, and 750 N s/m2, respectively. Other parameters of the local uniform nonlinear foundation are listed in Table 2. From Fig. 7, the second primary resonance regions shift to the lower frequency regions with the increase of the viscous damping. Meanwhile, the jumping down of the amplitude–frequency response curves gradually disappears in the above process. The increase of the viscous damping of the local uniform nonlinear foundation has a beneficial effect on the vibration suppression at both ends of the beam structure. Additionally, there is an unstable region in the amplitude–frequency curves of the beam structure under CI = 25 N s/m2. To further study the vibration state of the unstable region, the phase diagram and Poincaré points of the targeted region are plotted. When the excitation frequency is 36 Hz, the Poincaré points form a closed curve and the phase path stays stable. It can be concluded that the vibration state at 36 Hz is in the quasi-periodic state.

Fig. 7
Fig. 7
Close modal

Third, the influence of the distributed mass of the local uniform nonlinear foundation on the amplitude–frequency response curves at both ends of the beam structure is studied. Amplitude–concentrated-mass response curves at both ends of the beam structure under different distributed masses are plotted in Fig. 8, where the distributed masses of the local uniform nonlinear foundation are set as 0.25 kg/m, 0.75 kg/m, 1.25 kg/m, 1.75 kg/m, and 2.25 kg/m, respectively. Other parameters of the local uniform nonlinear foundation are listed in Table 2. From Fig. 8, the second primary resonance regions shift to the lower frequency regions with the increase of the distributed mass. The jumping down of the amplitude–frequency response curves always exists under different distributed masses. Additionally, decreasing the distributed mass of the local uniform nonlinear foundation beneficial affects the vibration suppression at both ends of the beam structure.

Fig. 8
Fig. 8
Close modal

Fourth, the influence of the center position of the local uniform nonlinear foundation on the amplitude–frequency response curves at both ends of the beam structure is studied. Amplitude–center-position response curves at both ends of the beam structure under different center positions are plotted in Fig. 9, where the center positions of the local uniform nonlinear foundation are set as 0.1 m, 0.2 m, 0.3 m, and 0.4 m, respectively. Other parameters of the local uniform nonlinear foundation are listed in Table 2. From Fig. 9, the vibration of the first primary resonance region in the amplitude–frequency response curve at the softer side is suppressed as the nonlinear foundation reaches the softer side of the beam structure. The vibration of the second primary resonance region in amplitude–frequency response curves at both ends is suppressed as the nonlinear foundation reaches the stiffer side of the beam structure. The jumping down of the amplitude–frequency response curves appears when the center positions of the nonlinear foundation are 0.1 m and 0.2 m. In contrast, the jumping down of the amplitude–frequency response curves nearly disappears when the center positions of the nonlinear foundation are 0.3 m and 0.4 m.

Fig. 9
Fig. 9
Close modal

In summary, parameters of the local uniform nonlinear foundation significantly influence the amplitude–frequency responses at both ends of the generally restrained axially loaded beam structure. Suitable parameters of the local uniform nonlinear foundation have a beneficial effect on vibration suppression at both ends of the beam structure under the forward sweeping way.

### 3.4 Influence of the Local Uniform Nonlinear Foundation on Single-Frequency Responses.

This section studies the influence of local uniform nonlinear foundation on dynamic responses at both ends of the beam structure under a single-frequency excitation. The excitation frequency is set as 136 Hz in this section. Other parameters of the harmonic point excitation are the same as those used in Sec. 3.1. Parameters of the initial values of the Galerkin truncated method are the same as those used in Sec. 3.1. Parameters of the beam structure and boundary conditions are listed in Table 1.

First, the influence of the nonlinear stiffness on the dynamic responses at both ends of the beam structure under a single-frequency excitation is studied. The nonlinear stiffness varies from 2.5 × 109 N/m4 to 2.5 × 1014 N/m4. The center position of the nonlinear foundation is set as xI = 0.2 m. The viscous damping of the nonlinear foundation is set as CI = 125 N s/m2. Other parameters of the nonlinear foundation are listed in Table 2. Stable steady-state amplitude–nonlinear-stiffness response curves at both ends of the beam structure under 115 Hz are plotted in Fig. 10. From Fig. 10, the nonlinear stiffness has a significant influence on the dynamic responses of the beam structure under 115 Hz. There are critical values of the nonlinear stiffness on amplitude–nonlinear-stiffness response curves. As the nonlinear stiffness exceeds its critical value, the vibration state of the beam structure changes greatly. For the parameters utilized in this part, the critical values of the nonlinear stiffness are 8.82 × 1013 N/m4 and 1.18 × 1014 N/m4. The unstable region in amplitude–nonlinear-stiffness response curves at both ends of the beam structure ranges from 8.82 × 1013 N/m4 to 2.5 × 1014 N/m4. To further investigate the vibration state in unstable regions, phase diagrams of the targeted regions are plotted. The Poincaré points are also plotted in the phase diagrams. From phase diagrams, when the nonlinear stiffness ranges from 8.82 × 1013 N/m4 to 1.18 × 1014 N/m4, the phase path stays stable and Poincaré points form a closed curve. The vibration state of the corresponding unstable region is quasi-periodic. When the nonlinear stiffness ranges from 1.18 × 1014 N/m4 to 2.5 × 1014 N/m4, the phase path stays stable and there are two Poincaré points. The vibration state of such an unstable region is in the double-periodic state.

Fig. 10
Fig. 10
Close modal

Second, the influence of the viscous damping of the nonlinear foundation on the dynamic responses at both ends of the beam structure under a single-frequency excitation is studied. The viscous damping of the nonlinear foundation varies from 25 N s/m2 to 1250 N s/m2. The center position of the nonlinear foundation is set as xI = 0.2 m. The nonlinear stiffness is set as knI = 1014 N/m4. Other parameters of the nonlinear foundation are listed in Table 2. Stable steady-state amplitude–viscous-damping response curves at both ends of the beam structure under 115 Hz are plotted in Fig. 11. From this figure, the viscous damping of the nonlinear foundation has a significant influence on the dynamic responses of beam structure at 115 Hz. There are three critical values of the viscous damping on the amplitude–viscous-damping responses curves. As the viscous damping of the nonlinear foundation exceeds its critical value, the vibration state of the beam structure varies greatly. For the parameters applied in this part, the critical values of the viscous damping are 330 N s/m2, 492 N s/m2, and 1127 N s/m2. Unstable regions in the amplitude–viscous-damping response curves at both ends of the beam structure range from 25 N s/m2 to 330 N s/m2 and 492 N s/m2 to 1127 N s/m2. Additionally, there is a region that has multiple amplitudes. To further study the vibration state of the above unstable regions, phase diagrams of the targeted regions are plotted. Poincaré points are also plotted in these phase diagrams. For the unstable region, the phase path and Poincaré points form a closed curve. The vibration state of the unstable regions in Fig. 11 is in the quasi-periodic state. For the region has multiple amplitudes, there are two Poincare points in each phase diagram, which suggests that its vibration state is in the double-periodic state.

Fig. 11
Fig. 11
Close modal

Third, the influence of the distributed mass of the nonlinear foundation on the dynamic responses at both ends of the beam structure under a single-frequency excitation is studied. The distributed mass of the nonlinear foundation varies from 0.25 kg/m to 1.25 kg/m. The viscous damping of the nonlinear foundation is set as CI = 125 N s/m. The center position of the nonlinear foundation is set as xI = 0.2 m. The nonlinear stiffness is set as knI = 1014 N/m4. Other parameters of the nonlinear foundation are listed in Table 2. Stable steady-state amplitude–concentrated-mass response curves at both ends of the beam structure under 115 Hz are plotted in Fig. 12. From Fig. 12, the distributed mass of the nonlinear foundation significantly influences the dynamic responses of the beam structure under 115 Hz. There are critical values of the distributed mass on the amplitude–concentrated-mass responses curves. As the distributed mass of the nonlinear foundation exceeds its critical value, the vibration state of the beam structure varies greatly. For the parameters employed in this part, the critical values of the distributed mass are 0.88 kg/m and 1.15 kg/m. There are two unstable regions in the amplitude–concentrated-mass response curves at both ends of the beam structure. The ranges of such unstable regions are 0.25 kg/m to 0.88 kg/m and 1.15 kg/m to 1.25 kg/m. To further study the vibration state of the beam structure, phase diagrams and Poincaré points of the targeted regions are plotted. For the unstable regions in Fig. 12, the phase path stays stable and the Poincaré points form a closed curve. The vibration state of the unstable regions is in the quasi-periodic state.

Fig. 12
Fig. 12
Close modal

Fourth, the influence of the center position of the nonlinear foundation on the dynamic responses at both ends of the beam structure under a single-frequency excitation is studied. The center position of the nonlinear foundation varies from 0.1 m to 0.48 m. The viscous damping of the nonlinear foundation is set as CI = 250 N s/m2. The nonlinear stiffness is set as knI = 2.5 × 1012 N/m4. Other parameters of the nonlinear foundation are listed in Table 2. Stable steady-state amplitude–center-position response curves at both ends of the beam structure under 115 Hz are plotted in Fig. 13. From this figure, the dynamic responses of beam structure under 115 Hz are significantly affected by the center position of the nonlinear foundation. There are critical values of the center position on the amplitude–center-position responses curves. For the parameters employed in this part, the critical values of the center position are 0.41 m and 0.466 m. There are two unstable regions in the amplitude–center-position response curves at both ends of the beam structure. The ranges of such unstable regions are 0.41 m to 0.466 m and 0.467 m to 0.48 m. To further study the vibration state of such unstable regions, phase diagrams and Poincaré points of the targeted regions are plotted. For the unstable regions ranging from 0.41 m to 0.466 m, there are many Poincaré points in each phase diagram, where such Poincaré points are disordered. Therefore, the vibration state of such an unstable region is the chaotic state. For the unstable regions ranging from 0.467 m to 0.48 m, the phase path stays stable and Poincaré points tend to form a closed curve, which suggests that the vibration state of such an unstable region is in the quasi-periodic state.

Fig. 13
Fig. 13
Close modal

In summary, parameters of the local uniform nonlinear foundation have a significant influence on the dynamic responses at both ends of the beam structure under a single-frequency excitation. The complex dynamic behavior of the beam structure is motivated when the local uniform nonlinear foundation is in certain parameters. Suitable parameters of the local uniform nonlinear foundation beneficially influence the vibration suppression at both ends of the beam structure under a single-frequency excitation.

## 4 Conclusions

This study establishes the nonlinear vibration analysis model of a generally restrained axially loaded beam structure with a local uniform nonlinear foundation. The dynamic behavior of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation is predicted through the Galerkin truncated method, where the stability of the Galerkin truncated method is studied. The harmonic balance method is employed to verify the dynamic behavior predicted by the Galerkin truncated method. On this basis, the influence of the sweeping ways and the local uniform nonlinear foundation on the dynamic responses of the beam structure is investigated. For the generally restrained axially loaded beam structure with a local uniform nonlinear foundation studied in this work, some conclusions are drawn as follows.

1. Residual equations of the generally restrained axially loaded beam structure with a local uniform nonlinear foundation are derived and arranged into a matrix form, from which the nonlinear dynamic behavior of such a beam structure can be directly obtained in conjunction with the numerical method.

2. The nonlinear vibration analysis model of beam structure established by the Galerkin truncated method is verified through the harmonic balance method. A 4-term truncated number guarantees the stability of the Galerkin truncated in predicting the dynamic behavior of beam structure in this study.

3. Dynamic responses of a generally restrained axially loaded beam structure with a local uniform nonlinear foundation are sensitive to its calculation initial values. The increase of nonlinear stiffness enhances the difference between dynamic responses obtained through the forward and reverse sweeping way. The nonlinear stiffness motivates the complex dynamic responses of the beam structure.

4. The local uniform nonlinear foundation significantly influences the amplitude–frequency responses at both ends of the generally restrained axially loaded beam structure. For the simulation parameters employed in this study, suitable parameters of the local uniform nonlinear foundation can effectively suppress the vibration level at both ends of the beam structure under the forward sweeping way.

5. Dynamic behavior of the beam structure under a single-frequency excitation is significantly influenced by the parameters of the local uniform nonlinear foundation. For single-frequency responses, suitable parameters of the local uniform nonlinear foundation effectively transform the vibration state of the beam structure. On the other hand, suitable parameters of the local uniform nonlinear foundation beneficially influence the vibration suppression at both ends of the beam structure.

## Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant No. 11972125).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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