Example. On the steel frame of I-beams №. 24 there are two loads with masses m = 1000 kg and a working engine with N = 200 rev/min (rpm) (Fig. 3.6).The engine creates a vibration load, the maximum value of which is P0 = 100 N. Perform a frame calculation on the effect of the vertical component of the vibration load P0sinθt, considering the weights as point loads.
Own weight of the engine and frame rods neglected.
Solution:
1. Calculation of self oscillations
If we ignore the longitudinal vibrations of the rods, then the position of the two point masses of this frame will be determined by two independent parameters y1, y2 (Fig. a).Therefore, the number of dynamic degree of freedom for these system is two
Choose positive directions of movement and generalized forces (Fig.a, b). In the direction of the first degree of freedom, both masses oscillate, and in the direction of the second degree – only one of the masses.
Therefore, m1=m+m=2m, m2=m, and the mass matrix of the system will be
To determine the flexibility matrix δ, we consider two unit loadings of the system
(Fig. c, d) and in each of them we drawing the diagrams of bending moments
Calculate the elements of the flexibility matrix
In a statically indeterminate system, they can be calculated in two ways:
To determine the forms of self oscillations, we write the system of equations
in the usual form:
2. Check for resonance
Determine the circular frequency of the vibration force:
θ/ω1=20,93/19,6=1,07 ; θ/ω2=20,93/30,75=0,7
3. Dynamic strength test
Determine the vector of the amplitude values of the vibration load:
We write the canonical equations of the calculation of the vibration load
4. Check dynamic stiffness
Maximum mass movements during oscillations:
From tab. it follows that if a person is constantly near the 2nd mass during work (at the end of the cantilever part of the frame),then the oscillation amplitude will exceed the allowable (at a frequency f =3.33 Hz, the permissible amplitude a ≈ 1 mm)
