Question

This is a vibrations question. Please only answer it if you have excellent knowledge of vibrations. I will leave feedback that matched the quality and accuracy of your answer. Thank you.

This is a vibrations question. Th image below shows a bifilar pendulum. The length of the strings is given as H. The length of the rod is given as 2H. There is a mass (which is idendical to the rod mass) located in the centre of the rod as a point mass. When the bifilar pendulum swings it has 3 modes. H H 2H Although the swing of the bifilar pendulum has 3 modes, only 2 of them are important 1. Provide the equations for each period of the oscillation for the 2 modes that are important.

Answer 1

One mode will be in the plane of rod which will be torsional or twisting in nature. The other mode will be in the plane of rod and the strings which will be swaying in the shape.

For the twisting mode, the equilibrium equations are written first and then the tension in each string is determined. Torque balancing equations about the center of rod at O is then written. The equation is simplified with certain assumptions and the natural frequency for that particular mode is obtained. Once natural frequency is known, the time period can be known.

For the second mode, the rod and the string will move on the same plane. As the strings will move an angle θ from the horizontal, the rod will also twist by an angle θ about its center point. The equations balancing the torques are then written and the natural frequency is then obtained. Once the natural frequency is known, the time period can be obtained.

Equilibrium Equaticut (Rad muam 25 = ang + mg Center Point Mann) To ing. TA d e ang Tsind ming Taking memast about 0. Io + (Sin 4) + TT Sin 4) = 0 Tö + (275) H=0 Sind tan = => d= Hlau H d = lisina - Hsin Io + (2T Sino) = 0 d = HO) For Small e for Spacall o Sin Oo H(O) = Sind (e) ë + (2009) - Siu d = 0 L = 2H om (2H) + (20g H) Io + (27H)=0 22. I: → for Rool 8 + (6) -> From Equilibrium Equation hr Natural ܀ 8 tj7@ܐ ܕܪܗܝ (27H) o=0 frequency in swaying mene Time Period in 12 öt zugh o=0 140H² Sherrying Mode) 12. är (691H) o -]1914) → (Natural Prequency in Torsional Mode) 26 Time Peniel in Tohatual JOW) Mode) T= 20