The beam is divided into two parts at a concentrated mass point, as shown in Figure 1. The lengths of the two beam segments are denoted as a and b. Both beams have consistent density ρ, bending stiffness EI, and a concentrated mass M.

Assume that the two beams have different vibration mode functions, denoted as Y ₁ x) and Y ₂ x), as shown in Eq. 6 and Eq. 7). Y 1 x = C 1 ⁡ cosh ⁡ λ x + C 2 ⁡ sinh ⁡ λ x + C 3 ⁡ cos ⁡ λ x + C 4 ⁡ sinh ⁡ λ x (6) Y 2 x = D 1 ⁡ cosh ⁡ λ x + D 2 ⁡ sinh ⁡ λ x + D 3 ⁡ cos ⁡ λ x + D 3 ⁡ sinh ⁡ λ x (7)

Each mode function involves four parameters to be determined. The presence of concentrated mass points influences the boundary conditions of the two equations. Taking a cantilever beam as an example (as shown in Figure 2), the boundary conditions are expressed as Eqs 8–15:

1) At the clamped support, both the deflection and the rotation of the beam are zero. The boundary conditions for the fixed end are written as Eq. 8 and Eq. 9:

Bring the boundary conditions into Eq. 6 and Eq. 7 to get Eq. 16, and the final coefficient matrix is given as shown in Eq. 17. C 1 + C 3 = 0 C 2 + C 4 = 0 C 1 cosh ⁡ λ a − ⁡ cos ⁡ λ a + C 2 sinh ⁡ λ a − ⁡ sin ⁡ λ a − D 1 − D 3 = 0 C 1 λ sinh ⁡ λ a + ⁡ sin ⁡ λ a + C 2 λ cosh ⁡ λ a − ⁡ cos ⁡ λ a − D 2 − D 4 = 0 C 1 λ 2 cosh ⁡ λ a + ⁡ cos ⁡ λ a + C 2 λ 2 sinh ⁡ λ a + ⁡ sin ⁡ λ a − D 1 + D 3 = 0 C 1 E I λ 3 sinh ⁡ λ a − ⁡ sin ⁡ λ a − M λ 2 cosh ⁡ λ a + ⁡ cos ⁡ λ a + C 1 E I λ 3 cosh ⁡ λ a + ⁡ cos ⁡ λ a − M λ 2 sinh ⁡ λ a + ⁡ sin ⁡ λ a − D 2 + D 4 = 0 D 1 ⁡ cosh ⁡ λ b + D 2 ⁡ sinh ⁡ λ b − D 3 ⁡ cos ⁡ λ b − D 4 ⁡ sin ⁡ λ b = 0 D 1 ⁡ sin ⁡ h λ b + D 2 ⁡ cosh ⁡ λ b + D 3 ⁡ sin ⁡ λ b − D 4 ⁡ cos ⁡ λ b = 0 (16) A = 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 cosh ⁡ λ b sinh ⁡ λ b − ⁡ cos ⁡ λ b − ⁡ sin ⁡ λ b 0 0 0 0 sinh ⁡ λ b cosh ⁡ λ b sin ⁡ λ b − ⁡ cos ⁡ λ b cosh ⁡ λ a sinh ⁡ λ a cos ⁡ λ a sin ⁡ λ a − 1 0 − 1 0 λ ⁡ sinh ⁡ λ a λ ⁡ cos ⁡ λ a − λ ⁡ sin ⁡ λ a λ ⁡ cos ⁡ λ a 0 − 1 0 − 1 λ 2 ⁡ cosh ⁡ λ a λ 2 ⁡ cos ⁡ λ a − λ 2 ⁡ cos ⁡ λ a − λ 2 ⁡ cos ⁡ λ a − 1 0 1 0 E I λ 3 ⁡ sinh ⁡ λ a − M λ 2 ⁡ cosh ⁡ λ a E I λ 3 ⁡ cosh ⁡ λ a − M λ 2 ⁡ sinh ⁡ λ a E I λ 3 ⁡ sin ⁡ λ x + M λ 2 ⁡ cosh ⁡ λ x E I λ 3 ⁡ cos ⁡ λ a + M λ 2 ⁡ sin ⁡ λ a 0 − E I λ 2 0 E I λ 2 (17)

The coefficient matrix Eq. (17) in row-column form must be zero, resulting in the characteristic equation of a simply supported beam with non-uniform mass. det A = 0 (18)

Organize the determinant and get Eq. 19 − M 4 E I = λ 1 + ⁡ cosh ⁡ λ l ⁡ cos ⁡ λ l sinh ⁡ λ a + ⁡ sin ⁡ λ a [ sinh ⁡ λ l − ⁡ cosh ⁡ λ l cos ⁡ λ b + ⁡ sin ⁡ λ b + sin ⁡ λ l − ⁡ cos ⁡ λ l cosh ⁡ λ b + ⁡ sinh ⁡ λ b (19)

The same method can be used to derive the eigenvalue matrix equations for simply supported beams and clamped supported beams with a concentrated mass point.

In the simply supported beam model depicted in Figure 3, the boundary conditions at the clamped end and the positions of the concentrated mass point are consistent with those of the cantilever beam. The distinction lies in the boundary conditions at the simply supported end, where deflection and bending moment are both zero, as illustrated in Eq. 20 and 21. Deflection : Y 2 b = 0 (20) Bending moment : Y 2 ″ b = 0 (21)

By introducing the boundary conditions into Eq. 6 and 7 is obtained, and the final coefficient matrix is given as shown in Eq. 22. A = 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 cosh ⁡ λ b sinh ⁡ λ b cos ⁡ λ b sin ⁡ λ b 0 0 0 0 λ 2 ⁡ cosh ⁡ λ b sinh ⁡ λ b − λ 2 ⁡ cos ⁡ λ b − λ 2 ⁡ sin ⁡ λ b cosh ⁡ λ a sinh ⁡ λ a cos ⁡ λ a sin ⁡ λ a − 1 0 − 1 0 λ ⁡ sinh ⁡ λ a λ ⁡ cos ⁡ λ a − λ ⁡ sin ⁡ λ a λ ⁡ cos ⁡ λ a 0 − 1 0 − 1 λ 2 ⁡ cosh ⁡ λ a λ 2 ⁡ cos ⁡ λ a − λ 2 ⁡ cos ⁡ λ a − λ 2 ⁡ cos ⁡ λ a − 1 0 1 0 E I λ 3 ⁡ sinh ⁡ λ a − M λ 2 ⁡ cosh ⁡ λ a E I λ 3 ⁡ cosh ⁡ λ a − M λ 2 ⁡ sinh ⁡ λ a E I λ 3 ⁡ sin ⁡ λ x + M λ 2 ⁡ cosh ⁡ λ x E I λ 3 ⁡ cos ⁡ λ a + M λ 2 ⁡ sin ⁡ λ a 0 − E I λ 2 0 E I λ 2 (22)

The coefficient matrix Eq. 22 in row-column form must be zero, resulting in the characteristic equation of a simply supported beam with non-uniform mass. det A = 0 (23)

In the clamped supported beam model depicted in Figure 4, the boundary conditions at the clamped end and the positions of the concentrated mass point are consistent with those of the cantilever beam. The distinction lies in the boundary conditions at the another clamped supported end. These conditions entail the deflection of 0 and the Rotation of 0, as illustrated in Eq. 24 and 25. Deflection : Y 2 b = 0 (24) Bending moment : Y 2 ′ b = 0 (25)

By introducing the boundary conditions into Eq. 6 and 7 is obtained, and the final coefficient matrix is given as shown in Eq. 26. A = 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 cosh ⁡ λ b sinh ⁡ λ b cos ⁡ λ b sin ⁡ λ b 0 0 0 0 λ ⁡ sinh ⁡ λ b λ ⁡ cosh ⁡ λ b − λ ⁡ sin ⁡ λ b λ ⁡ cos ⁡ λ b cosh ⁡ λ a sinh ⁡ λ a cos ⁡ λ a sin ⁡ λ a − 1 0 − 1 0 λ ⁡ sinh ⁡ λ a λ ⁡ cos ⁡ λ a − λ ⁡ sin ⁡ λ a λ ⁡ cos ⁡ λ a 0 − 1 0 − 1 λ 2 ⁡ cosh ⁡ λ a λ 2 ⁡ cos ⁡ λ a − λ 2 ⁡ cos ⁡ λ a − λ 2 ⁡ cos ⁡ λ a − 1 0 1 0 E I λ 3 ⁡ sinh ⁡ λ a − M λ 2 ⁡ cosh ⁡ λ a E I λ 3 ⁡ cosh ⁡ λ a − M λ 2 ⁡ sinh ⁡ λ a E I λ 3 ⁡ sin ⁡ λ x + M λ 2 ⁡ cosh ⁡ λ x E I λ 3 ⁡ cos ⁡ λ a + M λ 2 ⁡ sin ⁡ λ a 0 − E I λ 2 0 E I λ 2 (26)

The coefficient matrix Eq. 26 in row-column form must be zero, resulting in the characteristic equation of a simply supported beam with non-uniform mass. det A = 0 (27)

The characteristic Eq. 19 and (23 and Eq. 27 is a nonlinear function of the λ. The Newton-Raphson iteration method can be employed to solve for λ.

Firstly, while ensuring that the total mass and geometric dimensions remain unchanged, the non-uniform mass Bernoulli-Euler beam is equivalently transformed into a uniform mass Bernoulli-Euler beam. Subsequently, the natural frequency of the uniform mass Bernoulli-Euler beam is calculated to obtain an analytical solution. This analytical solution serves as the initial approximate solution (x ₀ ) to initiate the iterative process.

Updating the initial guess using the Newton-Raphson iteration formula Eq. 28. x n = x n − 1 − f x n − 1 f ′ x n − 1 (28)

Check whether the new guessed value satisfies the predetermined convergence criteria. If satisfied, consider the root of the equation found; if not, go back to the previous step and repeat the iteration process.

The internal filling of the fuel element is complex, and the fuel rod is divided into multiple sections. As shown in Figure 5, the beam model is segmented into N sections with concentrated mass points as endpoints. It is simplified into a Bernoulli-Euler beam with N-1 concentrated mass points, each with a concentrated mass value of m a .

As shown in Eq. (29), it is simplified to the free vibration equation of a Bernoulli-Euler beam with N-1 concentrated masses m a . ∂ 2 ∂ x 2 E I ∂ 2 y x , t ∂ x 2 + ∑ a = 1 N − 1 m a δ x − x a m ∂ 2 y x , t ∂ t 2 + ρ A ∂ 2 y x , t ∂ t 2 = 0 (29) where, EI is the bending stiffness of the beam section; A is the beam section area; ρ is density; δ is the Dirac function; x a m is the horizontal coordinate of a concentrated mass point;

The free vibration of the curved beam is a fourth-order partial differential equation. To solve the differential equation, the separation variable method can be employed resulting in Eq. (30): y x , t = φ ¯ x T ¯ t (30)

In the equation, φ ¯ x represents the shape of the vibration; T ¯ t represents the amplitude that varies with time. By substituting Eqs 16 into Eq. 15 and rearranging, Eq. 31 is obtained: E I ∂ 4 φ ¯ x ∂ x 4 + λ ¯ i ∑ a = 1 N − 1 m a δ x − x a m φ ¯ x + λ ¯ ρ A φ ¯ x = 0 (31)

In this equation, λ ¯ = ω 2 represents the eigenvalues of the Bernoulli-Euler beam with concentrated mass. Obtaining an analytical solution for the variable coefficient differential equation is relatively challenging.

Consider the Euler beam with concentrated mass as a new system obtained by adding concentrated mass to a uniform mass Euler beam through system modification. The primary mode functions and eigenvalues of this new system can be obtained by perturbation calculations using the mode functions of a uniform mass beam. φ ¯ i x = φ i x + ∆ φ i x (32) λ ¯ i = λ i + ∆ λ i (33) φ ¯ i x and λ ¯ i represent the i order mode function and eigenvalue of the new system beam with concentrated mass; φ i x and λ i represent the i order mode function and eigenvalue of the uniform mass beam; ∆ λ i is the ith order eigenvalue correction of the new system with a concentrated mass point relative to the uniform mass beam. ∆ φ i x is the correction of the primary mode function of the new system with a concentrated mass point relative to the uniform mass beam, and it is a linear combination of the other retained primary mode functions of the uniform mass beams, excluding φ i x . ∆ φ i x = ∑ j = 1 , j ≠ i n φ j x q j (34)

In the equation, q j are coefficients of the modal linear combination. As long as δ λ i and is q j obtained, there are η unknowns, and by using Eqs 32 and 33, the i order main modal function φ ¯ i x and eigenvalue λ ¯ i of the beam with concentrated masses can be obtained. According to the theory of dynamics, the equivalent beam in Eq. (34) has an infinite number of main modes. However, in practical calculations, taking a finite number of low-order modes for approximation in perturbation solution can achieve results with sufficient accuracy.

Substituting Eqs 32 and 33 into Eq. 29, and simplifying using Eq. 31, yields the expression for Eq. 35: E I ∂ 4 ∆ φ x ∂ x 4 − λ i ρ A ∆ φ i x + ∆ λ i ρ A φ i x + ∆ λ i ρ A ∆ φ i x − ∑ a = 1 N − 1 m a δ x − x a m × λ i φ i x + λ i ∆ φ i x + ∆ λ i φ i x + ∆ λ i ∆ φ i x = 0 (35)

Substitute Eq. 34 into Eq. 35, then multiply both sides by φ k x (k = 1,2, … , η), integrate along the length L of the beam, and simplify using the modal orthogonality of the equivalent beam, resulting in Eq. (36): ∆ λ i m k δ k i + ∆ m k i + ∆ λ i ∑ j = 1 , j ≠ i n m k δ k j + ∆ m k j q j + ∑ j = 1 , j ≠ i n λ i − λ j m k δ k j + λ i ∆ m k j q j = − λ i ∆ m k i (36)

Where m k = ∫ 0 L ρ A φ k 2 x d x (37) δ m k = ∫ 0 L ∑ a = 1 N − 1 m a δ x − x a m φ k x φ i x d x = 0 (38)

Eq. 37 and Eq. 38 can be directly obtained using numerical integration. Letting k = 1, 2, … , η in Eq. 36 yields η nonlinear algebraic equations involving unknowns ∆ λ i ​ and q j ​. After organization and simplification, these equations can be expressed in matrix form as Eq. (39): A + λ i B + λ i C + q i q = p (39)

Where A = m 1 λ i − λ 1 ⋯ ⋯ 0 ⋮ m 1 λ i − λ 2 ⋯ ⋮ ⋮ ⋮ ⋱ ⋮ 0 ⋯ ⋯ m η λ i − λ η η × η (40) B = ∆ m 11 ∆ m 12 ⋯ ∆ m 1 η ∆ m 21 ∆ m 22 ⋯ ∆ m 2 η ⋮ ⋮ ⋮ ⋮ ∆ m η 1 ∆ m η 2 ⋯ ∆ m η η η × η (41) C = δ m 11 + m 1 δ m 12 ⋯ 0 ⋯ δ m 1 η δ m 21 δ m 22 + m 2 ⋯ 0 ⋯ δ m 2 η ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ δ m i 1 δ m 11 ⋯ 0 ⋯ δ m i η ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ δ m η 1 δ m η 2 ⋯ 0 ⋯ δ m η η + m η η × η (42a) p = − λ i δ m 1 i − λ i δ m 2 i ⋮ − λ i δ m η i η × 1 (43a) q = q 1 q 2 ⋯ q η 1 × η (44a)

Where the i of q is q i = ∆ λ i λ i .This transforms the variable coefficient differential Eq. (29) into a nonlinear matrix Eq. 39. λ i represents the eigenvalues of the free vibration of a uniform mass beam. The study of the free vibration characteristics of uniform mass beams under different boundary conditions is well-established, with standard analytical solutions available.

The nonlinear matrix Eq. (39) can be solved using the Newton-Raphson method or intelligent algorithms. The Newton-Raphson method is highly dependent on the choice of initial values. Providing reasonable initial values can not only reduce the number of iterations but also achieve more accurate convergence results. Given the meaning of the various combination coefficients within the vector q , initial values for q are specified. q = 0 (42b)

The termination criteria for iteration can be adopted as follows: q i ε − q i ε − 1 q ε ≤ ξ (43b)

In Eq. 43b: the superscript ε denotes the number of equation iterations; ξ is the convergence error, ξ = 1 × 10 − 8 . After obtaining the unknown vector q , substitute it into Eq. 32 and 33 to obtain the i natural frequency and mode shape of the beam with concentrated masses.

By comparing with the natural frequency of the non-uniform mass fuel rod obtained through finite element analysis (FEA) in ANSYS, the correctness of the rapid method is verified.

The finite element model is modeled using beam3 elements. The Beam3 element is a uniaxial element capable of withstanding tension, compression, and bending. Each node of this element has three degrees of freedom: linear displacements along the x and y directions, as well as angular displacement about the Z-axis.

As shown in Figure 6, the finite element model is divided into 17 elements, featuring a concentrated mass point with a mass of M. The lengths of the beams on either side of the concentrated mass are a and b. The key parameters governing the behavior of the system are detailed in Table 1, providing essential insights into the structural dynamics.

There is a concentrated mass in the middle of the beam (x = 1) with a concentrated mass of 1.57 kg. Under this condition, the relevant parameters in Eq. (19) can be expressed as a = 1, b = 1, M = 1.57kg, EI = 1/6 × 103, and solved Eq. 19 to obtain λn. The calculation results were shown in Table 2.

The natural frequency is calculated by Eq.44b and 45. ω n = λ n 2 E I ρ S l 4 (44b) f n = λ n 2 2 π E I ρ S l 4 (45)

The first four natural frequencies were determined using Eq. 44b and 44and the results are presented in Table 3. The theoretical calculations closely align with the simulation results, confirming the accuracy of the formula.

To evaluate the sensitivity of the formula to variations in the weight and position of the concentrated mass point, an additional analysis was performed. This involved altering both the location and weight of the concentrated mass point, and the corresponding results are provided in Table 4.

Eq. 19 is applicable to a cantilever Bernoulli–Euler beam with a single concentrated mass of varying weight. Across different parameter settings, the deviation between the theoretical formula and simulation results is consistently below 3%.

The parameters of the beam model remain consistent with Section 3.1. Three concentrated mass points are introduced onto the beam, dividing the beam model into four segments. Each concentrated mass has a mass of M, and each segment has a length of 0.25, as shown in Figure 7.

Table 5 presents the first four natural frequency values of the cantilever beam with concentrated mass points calculated using the finite element method and the proposed method. The study investigates the influence of the concentrated mass values on the computational accuracy. From Table 5, it can be observed that the number of concentrated mass points has a limited impact on the calculation error. However, as the weight of the concentrated mass increases, the calculation error also rises. Under various parameter settings, the deviation between the results obtained using Eq. 29 and the ANSYS simulation results is consistently below 3%.

To validate the impact of fixation methods on computational accuracy, three different fixation methods are considered (as shown in Figure 8): simply supported beam (CS), cantilever beam (CA) and clamped-supported beam (CC). The mass of concentrated mass point is 1.57/4 kg, and the length is L/4. The parameters of the beam model remain consistent with Section 3.1.

Table 6 lists the natural frequencies of non-uniform mass fuel rods under different boundary conditions. After segmenting the fuel rod based on internal filling, the analytical solutions obtained are generally similar to the results obtained through finite element calculations, with errors less than 3%. Among them, the fixed-supported beam has the smallest calculation error, while the cantilever beam has the largest. The stronger the constraint, the smaller the error in the analytical solution. First 4natural frequencies of segmented beams with different fixing methods.

Under various parameter settings, the deviation between the results obtained using the fast method and ANSYS simulation results consistently remains below 3%.

In summary, the model demonstrates accuracy in analyzing both single concentrated mass points and multi-concentrated mass points in the beam.

The fuel rod is one of the most important components in a nuclear reactor core. Typically, a fuel rod comprises enriched cylindrical ceramic pellets, gas plenums, and a reflector located at both ends of the fuel rod. These components are sealed within a stainless-steel cladding through the upper and lower end plugs. The length of the internal structure is tailored based on the specific service environment of the fuel rod in different reactors. Figure 9 illustrates the typical structure of fuel rods in a small lead-based reactor. The fuel rod has a length of 850 mm and a diameter of 9.3 mm. The fuel element includes upper and down end plugs, upper and down reflectors, the active zone, and a gas plenum, all enclosed by cladding. The materials, lengths, and masses of each section are detailed in Table 7.

As shown in Figure 10, finite element models for two different constraint methods are established. The finite element model is divided into 17 elements, and the linear density of each element is set based on the distribution of the filling material inside the fuel rod.

The natural frequency of the fuel rod is determined using the finite rod software ANSYS. In the model analysis, filler is introduced to the beam model in the form of attached mass. The first four natural frequencies of the fuel rod are then calculated using ANSYS, and the results are presented in Table 8.

To simplify the internal structure, the contents of the fuel rod are represented as concentrated mass points. This representation allows the fuel rod to be treated as a Bernoulli-Euler cantilever beam, with the concentrated mass points illustrated in Figure 11. The distribution of these concentrated mass points is based on the center-of-gravity of the internal fill.

Uniform mass distribution (Simplified Model 1): This assumes an even distribution of the filler’s mass within the fuel rod in the beam model. The natural frequency is then calculated using the vibration motion of a uniform mass cantilever beam.

Three concentrated mass points (Simplified Model 2): The internal filling of the fuel rod is organized into three concentrated mass points. The upper-end plug and the gas plenum are considered one concentrated mass point, the active zoom and the reflection are another concentrated mass point, and the lower end plug and the lower reflection form the third concentrated mass point. These mass points are distributed based on the center of gravity.

Five concentrated mass points (Simplified Model 3): Organize the internal filling of the fuel rod into 5 concentrated mass points, considering the upper plug, upper reflective, active zoom, lower end plug, and lower reflective as individual concentrated mass points. These mass points are distributed based on the center of gravity.

The natural frequencies of the three simplified models are presented in Table 9. In comparison with the simplified method using a uniform mass model (Simplified Model 1), the natural frequencies calculated using the new method (Simplified Model 2 and Simplified Model 3) exhibit increased accuracy. As depicted in Figure 12, the number of mass points influences the calculation accuracy with the new method. Setting a greater number of concentrated mass points results in more accurate calculations. For a model with five concentrated mass points, the error is less than 3% when compared with the calculation results obtained using ANSYS.

Moreover, this method demonstrates a faster calculation speed than ANSYS. By utilizing five concentrated mass points, the calculations equivalent to 17 nodes in ANSYS can be efficiently performed. In addition, when there are changes in the structure of the fuel rod, this method eliminates the need for remodeling; instead, it only requires the modification of relevant parameters to complete the modal analysis.

In general, the design lifespan for a fuel rod is typically set at 30 years. The fuel rod has a length of 850 mm and an uneven mass distribution. The weight of the gas pressurization chamber is significantly lighter than the other sections. By analyzing the position of the gas pressurization chamber and considering different constraint methods, the study investigates the impact of mass distribution on the natural frequencies of the fuel rod.

From Table 10 and Figure 14, In the case of fuel rods clamped at one end and simply supported at the other, this is the most commonly used fixing method in current reactors. Due to the asymmetric fixing method, when the gas chamber is near the simply supported end, there is a slight increase in the natural frequency of the fuel rod, with an increase of approximately 7%. This adjustment is made to reduce installation difficulty. In the case of fuel rods clamped at one end and free at the other, forming a cantilever beam structure, the sensitivity of the fuel rod to the position of the gas chamber is significant. When the gas chamber is close to the free end, there is a substantial increase in the natural frequency of the fuel rod, with a maximum increase of up to 17%.

As the lifespan of the fuel rod increases, it is necessary to increase the length of the gas plenum, to reduce the internal pressure of the fuel rod.

The effect of length on the natural frequency of the fuel rod is determined by varying the length of the gas plenum. The length of the gas plenum is 100mm, 150mm, 200mm, and 250 mm.Four cases with different length were established (In Table 11).

The first 4 natural frequencies for four scenarios were calculated using the fast computation model. The results are presented in Table 12 and Figure 15.

The research findings indicate a negative correlation between the natural frequency of the fuel rod and the length of the gas chamber. From Figure 15, in the case of clamped one end of the fuel rod and simply supporting the other, as the gas chamber length increases, the influence of the gas chamber position on the natural frequency gradually becomes more pronounced, especially when the gas chamber is close to the clamped end. The rate at which the natural frequency decreases with an increase in length becomes significantly more pronounced. In the scenario where one end of the fuel rod is clamped while the other end is free, this trend is even more noticeable. When the gas chamber length is near the clamped end, the decrease in the natural frequency of the fuel rod becomes more significant with an increase in the gas chamber length.

Therefore, if there is a need to increase the length of the plenum, it is recommended to place the plenum at the free end to enhance the first-order n*/atural frequency.

Compared to traditional fixing methods, the cantilever beam structure of the fuel rod exhibits a significantly lower natural frequency and higher sensitivity to mass, posing certain safety risks. However, the establishment of this fuel rod structure is designed to facilitate in-depth research for optimization.