Experiments on chaotic vibrations of a post-buckled cantilevered beam connected by a string to an axial spring

Abstract

Experimental results are presented on chaotic vibrations of a post-buckled cantilevered beam constrained by a string. The string is stretched between the top end of the beam and an axial spring. The axial spring consists of a leaf spring with an attached mass and is fixed on a base frame close to the clamped end of the beam. The length of the string is less than that of the beam. The beam is excited by lateral periodic acceleration. By increasing the attached mass on the axial spring, nonlinear responses of the beam are examined. Nonperiodic response is observed in a typical frequency region. The response is examined by the Fourier spectrum, maximum Lyapunov exponents, Poincaré projection, and principal component analysis. Predominant chaotic response is generated by the internal resonance with a frequency ratio of one-to-three. The fundamental mode and the second mode of vibration are strongly coupled in the chaotic response. When the attached mass of the axial spring is increased, the natural frequency of the axial spring approaches the region of chaotic response. Therefore, the number of vibration modes that contribute to the chaos increases. Furthermore, the Poincaré projection of the chaotic response shows more scattered figures.

Keywords

Beam chaotic vibrations flexible structure nonlinear vibrations vibration of continuous system

1. Introduction

Beams and strings are used as fundamental elements of thin-walled structures, such as artificial satellites and space vehicles. When a cantilevered beam is constrained by a string stretched from the clamped end to the top end, the beam is deformed to a curved configuration in a post-buckled state. Then, the rigidity of the cantilevered beam can be increased. Furthermore, the post-buckled cantilevered beam can be used as a simplified manipulator since the deflection of the beam can be easily controlled by the string. However, when the beam is subjected to periodic lateral acceleration, nonlinear vibrations of the beam with large amplitude are generated. In specific frequency ranges, chaotic vibrations are also generated. Therefore, it is of importance to investigate the nonlinear and chaotic vibrations of the post-buckled beam connected by the stretched string.

Nonlinear and chaotic vibrations of beams were studied by many researchers, including the authors. Nonlinear vibrations of buckled beams with both ends clamped were investigated by Tseng and Dugundi (1971). Occurrence of irregular snap-through response was mentioned. Chaotic vibrations of cantilevered beams with a two-well potential function were investigated by Holmes (1979) and Moon and Holmes (1979). Pezeshki and Dowell (1989) investigated chaotic responses of a buckled beam using the Lyapunov dimension to estimate the number of vibration modes in the chaos. Smelova and Dowell (1996) investigated the effects of higher modes on the chaotic motion of buckled beams. Azeez and Vakakis (2001) introduced the principal component analysis to estimate the contribution of vibration modes in the vibro-impact problem of acantilevered beam. The authors have been investigating nonlinear and chaotic vibrations of arches (Nagai, 1985, 1986), post-buckled clamped beams (Nagai, 1990), post-buckled beams with concentrated mass (Nagai andYamaguchi, 1994; Yamaguchi and Nagai, 1994) and post-buckled reinforced beams (Nagai et al., 1998). Recently, detailed experimental results have been obtained on chaotic vibrations of a post-buckled clamped beam constrained by an axial spring (Nagai et al., 2007). Furthermore, corresponding analysis has been conducted to investigate modal contribution to the chaotic responses (Maruyama et al., 2008). For chaotic vibrations of a cantilevered beam constrained by a stretched string, experimental results (Nagai et al., 1994) and analytical results (Nagai et al., 2008) have been presented by the authors. In the cantilevered beam constrained by the stretched string, the axial force and the shearing force of the beam change as the deflection of the beam is increased, owing to the geometrical nonlinear coupling between the beam and the string. When the string is sufficiently longer than that of the beam, the compressive force is applied to the beam by the string parallel to the base line of the beam, which corresponds to the Euler buckling. In the previous experiment (Nagai et al., 1994), the length of the string was equal to that of the cantilevered beam. When the length of the string is shorter than that of the beam, it is expected that the deformation of the beam and the displacement of the string couple more drastically and that multiple vibration modes are generated in the chaotic responses. Furthermore, because coupling between the string and beam cause movement of an axial spring, which generates tension of the string, the rigidity and mass of the axial spring may influence the responses of the beam. It is of importance to investigate the effects of the axial spring and its mass on the chaotic response for the practical use of the beam constrained by the string as a manipulator, for example.

In this paper, experimental results are presented on chaotic vibrations of a post-buckled cantilevered beam constrained by a string. The string is stretched between the top end of beam and an axial spring. The length of the string is selected less than the length of the beam. The beam is deformed to a post-buckled configuration by the stretched string. The beam on the base frame is excited by lateral periodic acceleration. By increasing the attached mass on the axial spring, nonlinear responses of the beam are examined.

2. Test beam and fixture

As shown in Figure 1, a thin phosphor-bronze beam with thickness h = 0.503 mm, length L = 90.2 mm, and breadth b = 30.0 mm is clamped at one end of the beam. Material properties of the beam are measured as the Young’s modulus E = 110 GPa and the mass density ρ = 8.78 × 10³ kg/m³. Two steel strings with mean diameter d = 0.73 mm and length L_s = 62 mm are stretched between the other end, that is, the top end, of the beam and axial springs. Each axial spring consists of a leaf spring with an attached mass and is fixed on a base frame close to the clamp end of the beam. By moving the screw of the sliders, the strings are stretched. Then, post-buckled deformation of the beam is obtained by the axial compressive force applied to the beam by the string. To prevent torsional motions of the beam, the two strings are connected using the pulleys; thus, the tensile force of both strings becomes constant. The tensile force of the strings, which corresponds to the compressive force of the beam, can be measured with the strain gauges pasted on the leaf springs. The mass of the fixture of the string (e.g. the pulleys and their fixture) is 2.3 × 10^–3 kg. The coordinate system is denoted by the x-axis and z-axis, in the axial direction and the lateral direction of the beam, respectively, where the direction of the z-axis corresponds to the direction of the gravity.

3. Vibration test apparatus and procedure of the experiment

As fundamental properties of the beam, linear natural frequencies and characteristics of nonlinear restoring force of the beam are measured. Applying periodic sound pressure on the beam, resonant response with infinitesimal small amplitude is measured by a laser displacement sensor. Natural frequencies of the beam are inspected with a spectrum analyzer. Natural modes of vibration are detected by scanning the sensor along the beam. Characteristics of nonlinear restoring force of the beam are obtained through static deflection by a static concentrated force. The laser displacement sensor and a load cell are used. The concentrated force is applied to the beam, by pressing the detection needle of the load cell to the beam. Then, the beam deflects to an equilibrium position. Thus, the characteristics of the nonlinear restoring force of the beam can be obtained.

Figure 2 shows a schematic diagram of the vibration test apparatus. The base frame of the beam, which is mounted on the vibration table, is shaken periodically with an electromagnetic exciter. Thus, periodic lateral acceleration can be applied on the beam. The excitation is provided by the devices numbered from 1 to 5. The exciter controller (1) (B&K 1050) generates a sinusoidal periodic signal. The periodic signal is amplified through the power amplifier (2) (B&K 2078). The vibration exciter (3) (B&K 4802 and 4818) drives the base frame with periodic acceleration. The acceleration pickup (4) (B&K 4371) fixed on the base frame detects the acceleration applied on the beam (5). The signal of acceleration is fed back to the controller (1). Thus, the peak amplitude of acceleration can be kept constant. The electromagnetic exciter can generate the maximum amplitude of periodic force 1780 N. The excitation frequency can be swept from 20 Hz to 10 kHz with the resolution 1.2 mHz. The lowest sweep speed is 1 mHz/s. Dynamical responses of the beam are measured with the instruments of the multiple laser displacement sensors (6–8) (Keyence LC2100, measuring range ±3 mm). To inspect modal patterns generated in the chaotic responses, three sets of laser displacement sensors (6) are arranged over the beam. The displacement of the beam relative to the base frame is detected with the displacement sensors (6 and 7). The sensors (6) measure the responses of the beam and the displacement of the frame simultaneously. The sensor (7) measures only the periodic displacement of the base frame. The controller (8) subtracts two signals. With this subtraction, the pure dynamic response w(τ) of the beam can be detected, where w and τ indicate the nondimensional deflection and time, respectively, which will be defined below. One of the sensors (6) is set on the sliding table (9) and the sensor travels along the beam. Thus, static deflection and the vibration modes of the beam can be inspected. Nonlinear frequency response curves of the beam are obtained by sweeping the excitation frequency. The time responses of the beam detected by the laser displacement sensors are transformed to the amplitude in a root-mean-square value w_rms with the digital voltmeter (10) (Advantest TR6841). Through the periodic acceleration detected with the accelerometer (4), the excitation frequency f of the periodic acceleration is counted with the digital frequency counter (11) (Advantest TR5822). The amplitude of the response w_rms and the excitation frequency f are transferred to the computer (12). The digital spectrum analyzer (13) (Advantest TR9405) records time responses of chaotic vibrations and transforms the responses to the Fourier spectra. To inspect contribution of multiple vibration modes to the chaotic responses, time responses of the beam at three positions on the beam and the strain of the leaf spring are recorded simultaneously with the multi-channel digital recorder 14 (Yokogawa DL750). By applying the principal component analysis (Nayfeh and Balachandran, 1995; Feeny and Kappagantu, 1998) to these responses, modal contribution to the chaotic responses is inspected. Furthermore, the chaotic time responses are transmitted to the computer (12) and the maximum Lyapunov exponents λ_max are calculated with the procedure proposed by Wolf et al. (1985). The Poincaré projection of the response is obtained by the following procedure. Dynamic displacement of the chaotic responses is transformed to velocity by the differentiation amplifier (15). The phase meter (16) (B&K 2971) detects the maximum point of the periodic acceleration of excitation and then the pulse oscillator (17) (NF Elec. Instr. 1930) sends a timing signal to the digital recorder (14) with a prescribed phase shift. Then, the set of the displacement w and the velocity w,_ωτ is sampled sequentially based on the sampling pulse once in every period of the excitation.

To discuss the results of the experiments, the following nondimensional notations are introduced based on the governing equation of the motion of the post-buckled beam, which is shown in the appendix:

(1)

In Equation (1), r = (I/A)^1/2 represents the radius of gyration of the cross section of the beam, Ω₀ = L^–2(EI/ρA)^1/2 is the coefficient corresponding to the lateral vibration of the beam. The symbols I and A are the moment of inertia of the cross section of the beam I = bh³/12 and the area of the cross section A = bh. Notation ξ is the nondimensional coordinate and w is the lateral displacement normalized by the beam thickness h. Notation n_s is the nondimensional tensile force of the string, where N_s represents the tensile force. Notation β₁ is the mass ratio of the attached mass at the top end M₁ to the mass of the beam ρAL. In this paper, the top mass M₁ is assumed to be the sum of the mass of the fixture of the string and the half of the mass of the string. Notation β₂ is the mass ratio of the equivalent mass of the axial spring M₂ to the mass of the beam ρAL. The equivalent mass of the axial spring M₂ is estimated from the measurement of the natural frequency of the axial spring and the rigidity of the leaf spring. Notation l is the ratio of length of the string to that of the beam. Notation Γ is the slender ratio of the beam. Notation k₁ and k₂ are the coefficients of the spring corresponding to the string and the axial spring, respectively, normalized by the equivalent spring coefficient of the beam in the axial deformation. When the characteristics of restoring force of the beam are examined, the static lateral deflection by the concentrated static force Q_st is measured. Notation q_st is the nondimensional static force applied at the position ξ = ξ₂ on the beam. Notations p_d, ω, and τ are the nondimensional force intensity of the periodic excitation, the nondimensional excitation frequency, and the nondimensional time, respectively. Notation p_s is the nondimensional static force.

In this experiment, the parameters β₁, Γ, k₁, k₂, and l are fixed as β₁ = 0.22, Γ = 1.6 × 10^{- 3}, k₁ = 7.7 × 10^{- 2}, k₂ = 3.5 × 10^{- 3}, l = 0.69, while β₂ is changed as β₂ = 0.21, 3.3, 52.

4. Results and discussion

4.1. Fundamental properties of the beam

Natural frequencies of the beam are measured by increasing the tensile force of the string. In Figure 3, the ordinate shows the tensile force n_s, while the abscissa denotes natural frequencies ω₁ and ω₂ of the lowest and the second modes of vibration, respectively.

As the tensile force of the string n_s is increased, that is, the beam is compressed, the natural frequency of the lowest vibration mode ω₁ gradually increases. In contrast, the natural frequency of the second vibration mode ω₂ decreases until it takes the minimum value ω₂ ≈ 11 at the tensile force n_scr ≈ 16.5. This tensile force n_scr ≈ 16.5 corresponds to the critical load of the buckling. When the tensile force of the string is increased further, the second natural frequency increases steeply, because the curvature of the beam is increased.

Figure 4 shows the post-buckled deformed configuration of the beam constrained by the stretched string under the tensile force n_s = 19. In the figure, the ordinate represents the deflection w, while the abscissa denotes the coordinate ξ. The post-buckled configuration of the beam is shown by the rigid line. The post-buckled configuration is similar to the second natural mode of vibration of a cantilevered beam, since the beam is constrained by the string, which is shorter than the beam. The configuration of the buckled deformation explains the reason why the natural frequency of the second vibration mode ω₂ takes the minimum value as the tensile force of the string is increased. In Figure 4, the deformed configuration of the beam without the constraint by the string is shown by the dash-and-dotted line. The static deflection of the beam due to the gravitational force is of the same order as the thickness of the beam. The static deflection and the initial deflection of the beam cause the finite values of the natural frequencies at the buckling load.

Fixing the beam to the post-buckled form with the tensile force of the string n_s = 19, the following results of the vibration test are obtained. Table 1 shows the natural frequencies and natural modes of bending vibration of the beam at the tensile force n_s = 19. In the table, ω_ia, ω_ib, and ω_ic (i = 1, 2) denote the natural frequencies of the ith mode of vibration at the equivalent mass of the axial spring β₂ = 0.21, 3.3, and 52, respectively. The vibration modes for the equivalent mass of the axial spring β₂ = 0.21, 3.3, and 52 are plotted by a circle, triangle, and square, respectively. The arrows in the vibration modes express the direction of deformation of the axial springs. The natural frequencies of the lowest and the second modes have the relation 3ω₁ ≈ ω₂. Therefore, internal resonance may occur in a specific condition of excitation. The lowest vibration mode of the beam is similar to the lowest vibration mode of a cantilevered beam without constraint by a string. When the top end of the beam deflects in the positive z-direction, the axial spring deforms in the negative x-direction. The second vibration mode is similar to the lowest vibration mode of a beam with clamped and simply supported ends. When the center of the beam deflects in the positive z-direction, the axial spring deforms in the positive x-direction. The direction of the deformation of the axial spring in the vibration modes depends on the buckled configuration of the beam.

Table 1.

Natural frequencies and natural modes of bending vibration of the post-buckled beam (n_s = 19)

	mode	1	2
Natural frequencies	β₂ = 0.21	Ѡ_1a = 4.31	Ѡ_2a = 14.1
	β₂ = 3.3	Ѡ_1b = 4.47	Ѡ_2b = 12.6
	β₂ = 52	Ѡ_1c = 4.28	Ѡ_2c = 13.8
Vibration modes

Under the tensile force of the string n_s = 19, the lowest natural frequencies of the axial spring ω_s1a = 47.8, ω_s1b = 11.6, and ω_s1c = 4.9 are measured for the equivalent mass β₂ = 0.21, 3.3, and 52, respectively, by impacting the axial spring with an impact hammer. As the equivalent mass of the axial spring is increased, the natural frequency of the axial spring decreases. It is noteworthy that the natural frequency of the axial spring ω_s1b = 11.6 is close to the natural frequency of the second mode of vibration of the beam ω_2b = 12.6 for the equivalent mass β₂ = 3.3. It is expected that when the equivalent mass is β₂ = 3.3, the second vibration mode of the beam is easily coupled with the vibration of the axial spring. As a result, the second natural frequency ω_2b of the beam for the equivalent mass β₂ = 3.3 is lower than the second natural frequencies ω_2a and ω_2c for β₂ = 0.21 and 52, respectively, in the post-buckled state, as shown in Figure 3. For the equivalent mass β₂ = 52, the natural frequency of the axial spring ω_s1c = 4.9 is close to the lowest natural frequency of the beam ω_1c = 4.28.

Figure 5 shows the characteristics of the nonlinear restoring force of the beam under the tensile force n_s = 19. In the figure, the ordinate denotes the concentrated static load at the top end of the beam, while the abscissa denotes the deflection of the beam at the positions ξ = 0.89 and ξ = 0.5. The static equilibrium point of the buckling state at each measuring point is selected as the origin of deflection. The curve at the position ξ = 0.89 exhibits the characteristics of the restoring force of the type of a softening-and-hardening spring, including negative gradient. The dotted line in the figure denotes the occurrence of the snap-through buckling. The center of the beam ξ = 0.5 deflects in the reverse direction of the defection at the top end.

4.2. Frequency response curves of the beam

Figure 6 shows the frequency response curves of the beam under the periodic acceleration with the amplitude p_d = 41.3 and the tensile force n_s = 19. The ordinate denotes the amplitude w_rms of the beam at ξ = 0.5 in the root-mean-square value, while the abscissa denotes the excitation frequency ω. Notation (i, j) denotes the type of resonance, in which i is a generated mode of vibration, while j denotes a type of resonance. For example, j = 1 indicates the principal resonance, while j = 1/2 is the sub-harmonic resonance of 1/2 order. Natural frequencies of the beam are also plotted on the abscissa.

Figure 6(a) shows the results of the equivalent mass β₂ = 0.21 and 3.3. In the result of β₂ = 0.21, when the excitation frequency ω is decreased from the higher range, the principal resonance of the second vibration mode (2,1) appears around ω ≈ ω_2a = 14.1 with large amplitude, and then the amplitude suddenly decreases to nonresonance response of small amplitude with the jump phenomenon. As the excitation frequency ω is decreased further, the principal resonance of the lowest vibration mode (1,1) appears around ω ≈ ω_1a = 4.31. Furthermore, a chaotic response is generated near the largest amplitude of the resonant response of (1,1), as marked by C₁. The response curves of the resonant responses (1,1) and (2,1) correspond to the characteristics of restoring force of a softening spring.

The response curve of β₂ = 3.3 is similar to that of β₂ = 0.21 in the resonant response of the principal resonance of the lowest vibration mode (1,1) and the chaotic response marked by C₂. However, the peak amplitude of the resonant response of the second mode of vibration is smaller than that of β₂ = 0.21. Furthermore, another peak of the response curve appears because of the resonance of the axial spring around ω ≈ ω_s1b.

Figure 6(b) shows the result of the frequency response curve for the equivalent mass of β₂ = 52 compared with the result of β₂ = 0.21. In the result of β₂ = 52, a large amplitude response is generated around ω ≈ 4.5, where the vibrations of the beam and the axial spring are coupled. A chaotic response is also generated in the case of β₂ = 52, as marked by C₃.

4.3. Time histories and Fourier spectra of chaotic responses

Figures 7(a) and (b) show the time history and the Fourier spectrum, respectively, of the chaotic responses at the excitation frequency ω = 3.73 and the equivalent mass β₂ = 0.21, ω = 3.75 and β₂ = 3.3, and ω = 3.75 and β₂ = 52. In Figure 7(a), the ordinate denotes the nondimensional deflection w at the position ξ = 0.5, while the abscissa denotes the nondimensional time τ/τ_e normalized by the period of excitation τ_e = 2π/ω. The amplitude of the chaotic response is irregularly modulated. In Figure 7(b), the ordinate denotes the amplitude A of the spectrum scaled by decibel, while the abscissa denotes the nondimensional Fourier frequencyω_sp. In the broad band spectrum, the dominant spike of the spectrum appears at the excitation frequency ω and at the triple of the excitation frequency 3ω, which are close to the natural frequencies ω_1a, ω_1b, ω_1c and ω_2a, ω_2b, ω_2c, respectively, of the lowest and second modes of vibration. Therefore, the chaotic response is generated satisfying the condition of internal resonance ω_1a:ω_2a ≈ 1:3. Owing to the coupling between the deflection of the beam and the displacement of the axial spring, the dominant peak of the spectrum also appears at the twice the excitation frequency 2ω.

4.4. The maximum Lyapunov exponent

Based on the time progresses of the chaotic responses, the maximum Lyapunov exponents are calculated by Wolf’s method. Figure 8 shows the maximum Lyapunov exponent λ_max related to the embedding dimension e. In the result of ω = 3.73 and β₂ = 0.21, the maximum Lyapunov exponent λ_max converges to λ_max ≈ 0.1 as the embedding dimension increases to e = 5. In the results of ω = 3.75, β₂ = 3.3 and ω = 3.75, β₂ = 52, the maximum Lyapunov exponent λ_max converges to λ_max ≈ 0.2 as the embedding dimension increases from e = 4 to e = 5 and e = 8, respectively. As the maximum Lyapunov exponent takes a positive value, these responses are confirmed to be chaos. The number of vibration modes that contribute to a chaotic response can be estimated from the half of the embedding dimension where the convergence of the maximum Lyapunov exponent is obtained. Therefore, two or three vibration modes contribute to the chaotic responses of β₂ = 0.21 and β₂ = 3.3. As the equivalent mass of the axial spring is increased to β₂ = 52, the number of vibration modes that contribute to the chaotic response increases to four. As the attached mass of the axial spring is increased, the natural frequency of the axial spring approaches the region of chaotic response. Thus, the number of vibration modes that contribute to the chaos increases.

4.5. Poincaré projections

Figure 9 shows the Poincaré projections of the chaotic responses. The responses of the deflection w and the velocity w,_ωτ are sampled 6000 points at the phase delay θ from the maximum amplitude of the excitation acceleration. Figure 9(a) shows the result of ω = 3.73 and β₂ = 0.21 at the phase delay θ = 0, π /3, 2π/3. The figure of projection has a dense layer of scattered points, which rotate clockwise accompanied by folding-and-stretching in the fractal figure as the phase angle shifts.

Figures 9(b) and (c) show the Poincaré projections of the responses of ω = 3.75, β₂ = 3.3 and ω = 3.75, β₂ = 52, respectively. As the equivalent mass of the axial spring is increased, the figure of projection becomes a more scattered one, which corresponds to the increase in the number of vibration modes that contribute to the chaos.

4.6. Principal component analysis

To inspect the contribution of multiple vibration modes to the chaotic responses, time responses of the beam at three positions of the beam and the strain of the leaf spring are recorded simultaneously with the multi-channel digital recorder (14). By applying principal component analysis (Nayfeh and Balachandran, 1995; Feeny and Kappagantu, 1998) to these responses, modal contributions to the chaotic responses are inspected. Figure 10 shows the contribution ratio in the chaotic responses of ω = 3.73, β₂ = 0.21, of ω = 3.75, β₂ = 3.3 and of ω = 3.75, β₂ = 52. The ordinate denotes the contribution ratio of each principal component, while the abscissa is the order of the contribution ratio. The modal pattern of each principal component is also illustrated in the figure. The arrows in the modal pattern denote the displacements of the two leaf springs. The largest principal component, which corresponds to the lowest vibration mode, dominates the contribution ratio by more than 90%. The second largest contribution corresponds to the second mode of vibration with the contribution ratio from 5% to 8%. The lowest and second vibration modes are dominated in the chaotic responses that are generated from the internal resonance. In the modal pattern calculated by the principal component analysis, the phase relation between the deflection of the beam and the displacement of the axial spring coincides with that of the natural modes of vibration. Although the third and forth principal components have a fairly small ratio of contribution, the contribution ratio of these higher principal components increases as the attached mass of the axial spring is increased.

5. Conclusions

Experimental results are presented on chaotic vibrations of a post-buckled cantilevered beam constrained by a string. The length of the string is selected less than the length of the beam. The main results are summarized as follows.

Predominant chaotic response is generated by the internal resonance with a frequency ratio of one-to-three. The fundamental mode and the second mode of vibration are strongly coupled in the chaotic response.

When the attached mass of the axial spring is increased, the natural frequency of the axial spring approaches the region of chaotic response. Therefore, the number of vibration modes that contribute to the chaos increases. Furthermore, the Poincaré projection of the chaotic response shows more scattered figures.

Footnotes

Appendix

The governing equation and the corresponding boundary conditions for the nonlinear vibration of the post-buckled beam connected by a string to an axial spring are given with the nondimensional notations as: (2)

(3)

(4)

(5)

(6)

In the above equations, a subscript following a comma stands for partial differentiation. The following nondimensional notations are introduced, adding to the notations in Equation (1): (7)

In the notation in Equation (7), ŵ and w₀ are the nondimensional quantities of deflection W and initial deflection W₀ normalized by the radius of gyration r, respectively. Results of the experiment are presented with the nondimensional deflection w = W/h normalized by the thickness of beam h. The notation ${\hat{n}}_{x}$ represents the nondimensional quantity of the axial force N_x acting on the cross section of the beam. The notations û_s and u_e are the nondimensional quantities of axial displacement U_s of the axial spring and the initial axial displacement U_e applied to the axial spring by the slider, respectively. Since measurement of the initial axial displacement is difficult, compression of the beam is controlled by the nondimensional tensile force of the string n_s in the experiment. The equations and the boundary conditions include the nonlinear effects of axial deformation due to large deflections of ŵ and w₀.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Conflict of interest None declared.

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