Application of the Harmonic Balance Method to Centrifugal Pendulum Vibration Absorbers

Abstract

The harmonic balance method (HBM) is a powerful analysis tool for nonlinear vibrating systems, provided that the forms of the nonlinearities of the system result in a manageable algebraic system of equations. The authors of Cochelin and Vergez (J Sound Vib 324(1):243–262, 2009) created a framework that modifies the structure of the equations of motion involving a wide variety of nonlinearities into a quadratic form, which then can be approximated with HBM with as many assumed harmonics the problem needs for a satisfactory accuracy. In this work, we employ this framework for the analysis of centrifugal pendulum vibration absorbers (CPVA). The crucial step of this framework is the recasting of the variables into the required form. It has been shown that the dimensionless equations of motion for point mass CPVAs with general paths fitted to a rigid rotor can be put into the quadratic polynomial form. Two benchmark problems with known dynamical characteristics are investigated and the results show that this approach provides a powerful tool for investigating steady-state responses of these absorber systems. This will be very beneficial for design evaluations of CPVA systems where parameter values do not allow for the application of perturbation methods and/or make direct simulations very time consuming.

Appendix: Variable Recasting

The procedure starts by inserting the coordinates and their derivatives of the original equations into the new unknown vector, Z.

$$ \displaystyle{ z_{1} =\nu } $$

(25.29)

$$ \displaystyle{ z_{2} =\nu ' } $$

(25.30)

$$ \displaystyle{ z_{(3j)} = s_{j} } $$

(25.31)

$$ \displaystyle{ z_{(3j+1)} = s_{j}' } $$

(25.32)

$$ \displaystyle{ z_{(3j+2)} = s_{j}'' } $$

(25.33)

The next step is to define variables to represent the radial position of the absorber mass and its first and second derivatives as shown

$$ \displaystyle{ z_{(3N+3j)} = r_{j}^{2}(s_{ j}) } $$

(25.34)

$$ \displaystyle{ z_{(3N+3j+1)} = \frac{d(r_{j}^{2}(s_{j}))} {ds_{j}} } $$

(25.35)

$$ \displaystyle{ z_{(3N+3j+2)} = \frac{d^{2}(r_{j}^{2}(s_{j}))} {ds_{j}^{2}}. } $$

(25.36)

Such that

$$ \displaystyle{ z_{(3N+3j)} = 1 -\tilde{ n}_{j}^{2}z_{ (3j)}^{2} +\phi _{ j}z_{(3j)}^{4} } $$

(25.37)

$$ \displaystyle{ z_{(3N+3j+1)} = -2\tilde{n}_{j}^{2}z_{ (3j)} + 4\phi _{j}z_{(3j)}^{3} } $$

(25.38)

$$ \displaystyle{ z_{(3N+3j+2)} = -2\tilde{n}_{j}^{2} + 12\phi _{ j}z_{(3j)}^{2}. } $$

(25.39)

In order to account for powers of absorber position and velocity coordinates that are higher than quadratic, another set of variables are defined as

$$ \displaystyle{ z_{(6N+2j+1)} = z_{(3j)}^{2} } $$

(25.40)

$$ \displaystyle{ z_{(6N+2j+2)} = z_{(3j+1)}^{2}. } $$

(25.41)

The most problematic terms in the equations of motion stem from the function g(s) and its derivative, as the definition of the former involves the square root of an expression [see Eq. (25.7)]. To handle this we define a new set of variables and form a set of quadratic algebraic equations as follows

$$ \displaystyle{ z_{(8N+2j+1)} = g_{j}(s_{j}) } $$

(25.42)

$$ \displaystyle{ z_{(8N+2j+2)} = \frac{d(g_{j}(s_{j}))} {s_{j}} } $$

(25.43)

to obtain

$$ \displaystyle{ z_{(8N+2j+1)}^{2} = z_{ (3N+3j)} -\frac{1} {4}z_{(3N+3j+1)}^{2} } $$

(25.44)

$$ \displaystyle{ z_{(8N+2j+2)}z_{(8N+2j+1)} = z_{(3N+3j)} -\frac{1} {2}z_{(3N+3j+1)}z_{(3N+3j+2)}. } $$

(25.45)

Using these definitions, we can express the terms in the equations of motion using yet another variable definition

$$ \displaystyle\begin{array}{rcl} z_{(10N+j+2)}& =& -\mu _{\mathrm{a},j}z_{(3j+1)}z_{(8N+2j+1)} + z_{(2)}\left (z_{(3N+3j)} + z_{(3j+1)}z_{(8N+2j+1)}\right ) \\ & & +z_{(1)}\big(z_{(3j+1)}z_{(3N+3j+1)} + z_{(3j+2)}z_{(8N+2j+1)} + z_{(6N+2j+2)}z_{(8N+2j+2)}\big){}\end{array} $$

(25.46)

Finally, as the last layer of the recasting process, we define

$$ \displaystyle{ z_{(11N+2j+1)} = z_{(3N+3j)} + z_{(3j+1)}z_{(8N+2j+1)} } $$

(25.47)

$$ \displaystyle{ z_{(11N+2j+2)} = z_{(3j+1)}z_{(3N+3j+1)}+z_{(3j+2)}z_{(8N+2j+1)}+z_{(6N+2j+2)}z_{(8N+2j+2)}. } $$

(25.48)