Cantilever beam natural frequency formula
If course, we're ignoring higher modes here. RE: Natural frequency of a cantilevered beam with weight attached. Indeed, for the end at x = 0 the displacement u z 0 tand the slope u ′ z 0 tof displacement profile must be zero. In order to achieve this, four support conditions of the cantilever have to be used. (20) must be calculated taking account the conditions of the cantilever. Afterward, the four unknown parameters C 1, C 2, C 3, C 4of Eq. (1)– (20) describe mathematically the modal analysis of the inverted pendulum. Moreover, noted the resulting force P z x tof the external dynamic loading. On this infinitesimal length, we notice the flexural moment M x t, the shear force Q x twith their differential increments, while the axial force Ν x tis ignored, because it does not affect the horizontal cantilever vibration along z-axis. The infinitesimal length of this part is the dx. In order to formulate of the motion equation of this beam, we consider an infinitesimal part of the vertical beam, at location xfrom the origin o, that has isolated by two very nearest parallel sections. Due to fact that the cantilever mass is continuously distributed, this inverted pendulum/beam has infinity number of degrees of freedom for vibration along the horizontal oz-axis. Next, we are examining such an inverted pendulum that possesses constant value of distributed mass along its height, as well as constant value of distributed section flexural stiffness ΕΙ y. Furthermore, according to Bernoulli Technical Bending Theory, this cantilever has section flexural stiffness ΕΙ y x, where in the special case of an uniform distribution of the stiffness it is given as ΕΙ y x = ΕΙ y, where Eis the material modulus of elasticity and Ι yis the section moment of inertia about y-axis ( Figure 1). The vertical inverted pendulum possesses a distributed mass m xper unit height, which in the special case of uniform distribution is given as m x = m ¯in tons per meter (tn/m). Vertical cantilever (inverted pendulum) with distributed mass and section flexural stiffness. On this infinitesimal length, we notice the flexural moment M x t, the shear force Q x twith their differential increments, while the axial force Ν x tis ignored, because it does not affect the horizontal beam vibration along z-axis. Due to fact that the vertical pendulum mass is continuously distributed, this pendulum/beam has infinity number of degrees of freedom for vibration along the horizontal oz-axis. Next, we are examining a such amphi-hinge vertical pendulum that possesses constant value of distributed mass along its height, as well as constant value of distributed section flexural stiffness ΕΙ y. Furthermore, according to Bernoulli Technical Bending Theory, the beam has section flexural stiffness ΕΙ y x, where in the special case of an uniform distribution of the stiffness it is given as ΕΙ y x = ΕΙ y, where Eis the material modulus of elasticity and Ι yis the section moment of inertia about y-axis ( Figure 1). The vertical pendulum possesses a distributed mass m xper unit height, which in the special case of uniform distribution is given as m x = m ¯in tons per meter (tn/m). Moreover, the way of instrumentation with a local network by three accelerometers is given via the above-mentioned three degrees of freedom.Īmphi-hinge vertical pendulum with distributed mass and section flexural stiffness. Furthermore, taking account the 3 × 3 mass matrix, it is possible to estimate the possible pendulum damages using a known technique of identification mode-shapes via records of response accelerations. Using the three DoF model, the first three fundamental frequencies of the real pendulum can be identified with very good accuracy.
#CANTILEVER BEAM NATURAL FREQUENCY FORMULA FREE#
Based on the free vibration of the above-mentioned pendulums according to partial differential equation, a mathematically equivalent three-degree of freedom system is given for each case, where its equivalent mass matrix is analytically formulated with reference on specific mass locations along the pendulum height. These vertical pendulums have infinity number of degree of freedoms. The first case of inverted pendulum refers to an amphi-hinge pendulum that possesses distributed mass and stiffness along its height, while the second case of inverted pendulum refers to an inverted pendulum with distributed mass and stiffness along its height. In the present article, an equivalent three degrees of freedom (DoF) system of two different cases of inverted pendulums is presented for each separated case.