中低频集中质量悬臂式簧片仪基频分析方法研究

Fundamental frequency analysis method for medium-low frequencyconcentrated mass cantilever reed instrument

簧片仪是一种中低频冲击响应谱测量装置,设计时可将其简化为端部附带集中质量的悬臂梁。簧片仪的振型方程中存在超越函数,对于等截面簧片仪来说,可使用二分法等数值方法求解,计算量较大,该方法不适用于复杂截面簧片仪基频设计。首先利用悬臂梁自由端受集中力的挠度公式,推导出其等效刚度等效单自由度的弹簧振子系统,利用弹簧振子相关频率公式解决求解集中质量等截面悬臂梁的基频问题,该方法与振型方程求解的基频对比,发现在10 Hz以下的中低频区域,可以很好的保证等截面悬臂梁的基频精度,大于10 Hz的中频段,误差随频率的升高而迅速增大。接下来,通过Mohr积分的方法,推导出等强度集中质量悬臂梁自由端的最大挠度,并给出其基频的设计公式,该方法与试验进行对比,发现误差与等截面悬臂梁有相同趋势,因此利用瑞利能量法修正设计公式中的质量参数,修正后与实验误差在5%左右。经理论与试验验证,所提出的簧片仪设计方法简单可行、计算结果可信。

Reed instrument is a measuring device for medium-low frequency impact response spectra.It can be simplified as a cantilever beam with concentrated mass at its free end during its design.There is a transcendental function in its vibration mode shape equation.For a reed instrument with uniform cross-section,this equation can be solved with the bisection method,but the numerical computation is time-consuming.This method is not suitable for fundamental frequency design of reed instruments with complex cross-section.Here,firstly,the deflection formula for a cantilever beam subjected to a concentrated force at its free end was used to derive the equivalent stiffness of a single-DOF spring-mass system equivalent to the beam.The relevant natural frequency formula of this single-DOF spring-mass system was proposed to solve the fundamental frequency problem for a uniform cross-section cantilever beam with a concentrated mass.Comparing the beam’s fundamental frequency calculated using the proposed method with that using the vibration mode shape equation,it was found that within the medium-low frequency range of less than 10 Hz,the fundamental frequency accuracy of the cantilever beam with uniform cross-section can be well kept;within the medium frequency range of larger than 10 Hz,error rapidly increases with increase in frequency;Mohr integral method was used to deduce the maximum deflection at free end of a uniform strength concentrated mass cantilever beam,and give the design formula of its fundamental frequency.Comparing its fundamental frequency data calculated with the proposed method to test data,it was found that the error has the same trend as that of the cantilever beam with uniform cross-section;Rayleigh energy method is used to modify mass parameters in the design formula,and the error between the modified data and test ones is about 5%.Through theoretical calculation and test verification,it was shown that the proposed design method for a reed instrument’s fundamental frequency is simple and feasible,and the computation results are believable.