TWO DEGREE OF FREEDOM SYSTEM. INTRODUCTION. Systems that require two independent coordinates to describe their motion; Two masses in the system X two possible types of motion of each mass. Example: motor pump system.
Figure 5.3: A two degree of freedom spring-mass-damper system
The application of Newton’s second law of motion to each of the masses gives the equations of motion:
Both equations can be written in matrix form as
where [m], [c], and [k] are called the mass, damping, and stiffness matrices, respectively, and are given by
And the displacement and force vectors are given respectively:
It can be seen that the matrices [m], [c], and [k] are all 2 x 2 matrices whose elements are known masses, damping coefficient and stiffnesses of the system, respectively.
where the superscript T denotes the transpose of the matrix.
x1(t = 0) = x1(0) and 1( t = 0) = 1(0),
x2(t = 0) = x2(0) and 2 (t = 0) = 2(0).
By setting F1(t) = F2(t) = 0, and damping disregarded, i.e., c1 = c2 = c3 = 0, and the equation of motion is reduced to:
Assuming that it is possible to have harmonic motion of m1 and m2 at the same frequency ωand the same phase angle Φ, we take the solutions as
Substituting into Eqs.(5.4) and (5.5),
Since Eq.(5.7)must be satisfied for all values of the time t, the terms between brackets must be zero. Thus,
which represent two simultaneous homogenous algebraic equations in the unknown X1 and X2. For trivial solution, i.e., X1 = X2 = 0, there is no solution. For a nontrivial solution, the determinant of the coefficients of X1 and X2 must be zero:
which is called the frequency or characteristic equation. Hence the roots are:
The roots are called natural frequenciesof the system.
To determine the values of X1 and X2, given ratio
The normal modes of vibration corresponding to ω12 and ω22 can be expressed, respectively, as
which are known as the modal vectorsof the system.
The free vibration solution or the motion in time can be expressed itself as
Where the constants , , and are determined by the initial conditions. The initial conditions are
The resulting motion can be obtained by a linear superposition of the two normal modes, Eq.(5.13)
Thus the components of the vector can be expressed as
where the unknown constants can be determined from the initial conditions:
Substituting into Eq.(5.15) leads to
The solution can be expressed as
from which we obtain the desired solution
Find the free vibration response of the system shown in Fig.5.3(a) with k1 = 30, k2 = 5, k3 = 0, m1 = 10, m2 = 1 and c1 = c2 = c3 = 0 for the initial conditions
Solution: For the given data, the eigenvalue problem, Eq.(5.8), becomes
By setting the determinant of the coefficient matrix in Eq.(E.1) to zero, we obtain the frequency equation,
from which the natural frequencies can be found as
The normal modes (or eigenvectors) are given by
The free vibration responses of the masses m1 and m2 are given by (see Eq.5.15):
By using the given initial conditions in Eqs.(E.6) and (E.7), we obtain
The solution of Eqs.(E.8) and (E.9) yields
while the solution of Eqs.(E.10) and (E.11) leads to
Equations (E.12) and (E.13) give
Thus the free vibration responses of m1 and m2 are given by
Figure 5.6: Torsional system with discs mounted on a shaft
Consider a torsional system as shown in Fig.5.6. The differential equations of rotational motion for the discs can be derived as
which upon rearrangement become
For the free vibration analysis of the system, Eq.(5.19) reduces to
Find the natural frequencies and mode shapes for the torsional system shown in Fig.5.7 for J1 = J0 , J2 = 2J0 and kt1 = kt2 = kt .
Solution: The differential equations of motion, Eq.(5.20), reduce to (with kt3 = 0, kt1 = kt2 = kt, J1 = J0 and J2 = 2J0):
Rearranging and substituting the harmonic solution:
gives the frequency equation:
The solution of Eq.(E.3) gives the natural frequencies
The amplitude ratios are given by
Equations (E.4) and (E.5) can also be obtained by substituting the following in Eqs.(5.10) and (5.11).
Generalized coordinatesare sets of n coordinates used to describe the configuration of the system.
From the free-body diagram shown in Fig.5.10a, with the positive values of the motion variables as indicated, the force equilibrium equation in the vertical direction can be written as
and the moment equation about C.G. can be expressed as
Eqs.(5.21) and (5.22) can be rearranged and written in matrix form as
The lathe rotates in the vertical plane and has vertical motion as well, unless k1l1 = k2l2. This is known as elastic or static coupling.
From Fig.5.10b, the equations of motion for translation and rotation can be written as
These equations can be rearranged and written in matrix form as
If , the system will have dynamic or inertiacoupling only.
Note the following characteristics of these systems:
Determine the principal coordinates for the spring-mass system shown in Fig.5.4.
Approach: Define two independent solutions as principal coordinates and express them in terms of the solutions x1(t) and x2(t).
The general motion of the system shown is
We define a new set of coordinates such that
Since the coordinates are harmonic functions, their corresponding equations of motion can be written as
From Eqs.(E.1) and (E.2), we can write
The solution of Eqs.(E.4) gives the principal coordinates:
The equations of motion of a general two degree of freedom system under external forces can be written as
Consider the external forces to be harmonic:
where ω is the forcing frequency. We can write the steady-state solutions as
Substitution of Eqs.(5.28) and (5.29) into Eq.(5.27) leads to
We defined as in section 3.5 the mechanical impedance Zre(iω) as
And write Eq.(5.30) as:
Eq.(5.32) can be solved to obtain:
where the inverse of the impedance matrix is given
Eqs.(5.33) and (5.34) lead to the solution
Find the steady-state response of system shown in Fig.5.13 when the mass m1 is excited by the force F1(t) = F10cosωt. Also, plot its frequency response curve.
The equations of motion of the system can be expressed as
We assume the solution to be as follows.
Eqs.(E.4) and (E.5) can be expressed as
Fig.5.14: Frequency response curves