Abstract:In the extensive modern applications, the low-frequency and heavy-load isolators are needed to reduce the vibration transmissions. The unique properties of nonlinear systems, such as jumping, bifurcation and chaos, provide new ideas for designing the new functional structures. Bistable system is a typical non-linear system, features highly static and low dynamic stiffness, which promises to realize a low-frequency isolator with ensuring heavy load capacity. However, more studies are necessary to clarify the sub-harmonic resonance and its generation process, parameter influences, vibration isolation characteristics of the bistable structure.By adopting the equivalent, analytical, numerical and experimental methods, we study the 1/2 sub-harmonic resonance, evolution process and its influence on the vibration isolation characteristics of the bistable structure in this paper. When the amplitude or nonlinear stiffness coefficient kn increases to a certain extent, 1/2 sub-harmonic resonance appears, where the response contains high-amplitude ω/2 component under the excitation frequency ω, so the energy is transferred from high frequency to low frequency. We study the bifurcation and varying processes of the fundamental and 1/2 sub-harmonic transmission by increasing the amplitude. At critical bifurcation amplitude, the sub-harmonic transmission rapidly increases from 0 to a large peak value. And then, it decreases gradually when the damping is absent. However, the peak value of 1/2 sub-harmonic does not cause the fundamental transmission to change suddenly. When considerable damping appears with the increase of the amplitude, 1/2 sub-harmonic does not always exist, instead, it follows an interesting “generation-enhancement-degeneration-disappearance” process. This process possesses great significance in applying the 1/2 sub-harmonic to vibration manipulation or avoiding the resonant enhancement induced by it. Moreover, in this process, both the peak frequency and the peak transmission of the bistable isolation system descend first. The optimal combination of the parameters can reduce the resonance frequency by 17.8% through increasing the driving amplitude. However, they jump to large values when 1/2 sub-harmonic plays a dominant role. Additionally, the negative stiffness k0 has a significant effect on the primary resonance characteristics: as |k0| increases under a specified excitation amplitude, the resonance peak shifts toward higher frequency and the transmission increases. Besides the main effect on the sub-harmonic resonance and the equilibrium point, the nonlinear coefficient kn also affects the peak and resonance frequency of the system, but the effect is much less than the influence caused by k0.Furthermore, the sub-harmonic resonances, bifurcations and vibration isolation characteristics of the bistable bulking beam structure are demonstrated experimentally. The experimental results show that: 1) the 1/2 sub-harmonic resonance can appear in a certain bandwidth and it is not monochromic; 2) the increase of the driving amplitude can reduce the transmission of the fundamental wave; 3) the transmission of 1/2 sub-harmonic jumps from 0 upward to a large value at a critical amplitude, and then it decreases gradually. The experimental results are consistent with the analytical and numerical results. The experiment also demonstrates the law of frequency shifting and the transmission reduction of peak values. Therefore, the appropriate increase of the amplitude can improve the vibration isolation capacity. However, sub-harmonic resonance will reduce the isolation effect. In practical engineering, the strong sub-harmonic resonance should be avoided in a nonlinear vibration isolation system. Keywords:bistable structure/ sub-harmonic resonance/ vibration isolation characteristics/ buckling beam
全文HTML
--> --> -->
2.双稳态屈曲梁结构动力学特性分析方法本文研究的对象如图1(a)所示, 主要部分为两个并列的双稳态圆弧. 双稳态圆弧可保证实际实验中加载配重后样件的稳定性. 该结构可视作两个非线性弹簧并联. 在结构振动的过程中, 两弹簧位移相同, 共同承受附加在结构上的质量. 所研究对象可简化为图1(b)所示的非线性弹簧振子模型, 为两组相同的弹簧振子结构并联. 屈曲梁几何参数为: 跨距l = 35 mm, 厚度t = 1 mm, 高度h = 5.25 mm, 宽度b = 56 mm. 双稳态屈曲梁结构试样使用3D打印的方法制备, 材料为TPU (热塑性聚氨酯弹性体橡胶), 其材料参数通过压缩实验测得: 密度ρ = 916.7 kg/m3, 弹性模量E = 17.9396 MPa, 泊松比ν = 0.385, 总配重M为0.93 Kg. 图 1 (a)双稳态屈曲梁试样; (b)弹簧振子模型 Figure1. (a) Prototype of bistable buckling beam; (b) spring oscillator structure.
本文采用MATLAB软件的Simulink模块对系统进行数值仿真分析. 针对系统的微分方程(1)建立系统的数值仿真模型, 如图2所示. 系统的激励u为正弦波, y = x + u为系统响应的绝对位移. 利用该模型, 可求解系统在给定激励下响应的数值解. 图 2 Simulink数值仿真模型 Figure2. Numerical simulation model in Simulink.
-->
3.1.双稳态系统的1/2次谐波共振
给定U = 0.01 mm, 该系统的数值仿真结果与解析结果对比如图3所示. 实数解析结果具有3个分支, 双稳态系统的非线性使传递率曲线弯向左侧, 与软刚度非线性系统类似; 0—29 Hz频段, 解析解有两个根, 系统稳定状态下的响应对应的一般为幅值较低的解. 仿真结果与解析解的分支1和分支2一致性较好, 然而仿真结果中传递率在58 Hz处出现奇异峰值, 此频率恰好为系统主共振频率f0的2倍, 因此可能产生了1/2次谐波共振现象. 但是根据(5)式得到的解析结果并没有预测出该奇异峰值, 原因在于以上理论没有考虑次谐波响应, 所以预测精度较低. 图 3 解析与数值分析结果对比 Figure3. Comparison between analytical and numerical results.
共振频率和共振峰峰值是双稳态系统振动特性的两个重要指标, 也是隔振应用中十分关注的两个变量. 前文分析表明, 在分析主共振时解析计算中可以不考虑次谐波的影响, 不同幅值下的数值和解析结果分别如图8中实线和点线所示. 在共振区域, 数值与解析结果趋势一致: 随激励幅值增加, 共振峰峰值降低, 共振点的位置向低频移动, 峰值传递率下降. 数值仿真结果还表明, 在一定带宽的激励下(图中为55—58 Hz), 均有1/2次谐波共振产生. 图 8 无阻尼条件下幅值变化对隔振特性的影响 Figure8. Influence of amplitude on vibration isolation characteristics without damping.
当考虑阻尼效应时, 上文所述共振频率偏移量会显著增加. 数值分析表明(图9), 随着激励幅值增加, 其共振频率从28 Hz降低到23 Hz, 频率前移了17.8%, 峰值传递率也随之下降. 但是激励幅值增加到一定程度(此系统为U > 1.05 mm之后), 系统的峰值频率(不一定是主共振)急剧跳变增大, 峰值传递率也跳变升高. 由图7(b)可知, 这一跳变点恰好对应于T12 = T1, 峰值频率和传递率升高的原因在于高幅值的1/2次谐波改变了系统的本质特性. 图 9 有阻尼条件下幅值变化对频率偏移(a)和共振峰峰值(b)的影响 Figure9. Influences of amplitude on frequency shifting (a) and the peaks of harmonic resonance (b) with damping.
为研究并验证该结构的1/2次谐波共振现象, 本文在实验中施加频率为55 Hz的正弦激励并研究系统的响应特性, 结果如图13所示. 图13(a)表明, 系统频域响应曲线除在55 Hz处有尖峰外, 在接近27.5 Hz处也有一个明显的峰值. 由此可知, 本文设计的双稳态结构确实可以产生显著的1/2次谐波共振响应, 时域响应曲线每2倍周期响应幅值被加强. 此外, 本文实验还表明, 1/2次谐波共振现象能在基波为40—60 Hz的宽带范围内观察到, 证明了1/2次谐波共振在较宽的带宽下均可产生. 图 13 频率为55 Hz的正弦激励信号下系统的响应 (a)频域响应; (b)时域响应 Figure13. Response under sinusoidal excitation signal with frequency of 55 Hz: (a) Frequency domain; (b) time domain.
为从实验中验证激励幅值对系统次谐波共振的影响规律, 本文以40 Hz为例开展实验研究, 结果如图14和图15所示. 图 14 激励幅值变化对1/2次谐波共振的影响 Figure14. Influence of excitation amplitude on the 1/2 sub-harmonic resonance.
图 15 系统40 Hz处的次谐波共振现象 (a) U = 1.251 mm时的响应和激励频谱; (b), (c)U = 1.351 mm时的响应和激励频谱; (d)U = 1.351 mm时的时域波形 Figure15. Sub-harmonic resonance phenomena at 40 Hz: (a) Response and excitation spectrum with U = 1.251 mm; (b), (c) response and excitation spectrum with U = 1.351 mm; (d) time-domain waveform with U = 1.351 mm.