Comparison of translational and rotational motion parameter characteristics.
This chapter will give an introduction to linear and nonlinear oscillators and will propose literature to this topic. Most importantly, hands on examples with numerical simulations are illustrating oscillations and resonance phenomena and where useful, also analytical methods to treat nonlinear behavior are given.
- parametric resonance
- autoparametric resonance
- nonlinear vibration
- Mathieu equation
- Hopf bifurcation
- Strutt diagram
- nonlinear natural frequency
- instability domain
- basepoint excited primary and secondary system
When a mechanical system has at least two vibrating components, the vibration of one of the components may influence the other component. This influence effect which might stabilize or destabilize the system is called autoparametric resonance. This chapter will introduce autoparametric resonance by examining hands on examples for such systems. In particular, basepoint excited systems are analyzed. Beside purely mechanical systems, also examples of an electrical system with two coupled resonators are investigated.
There are three main types of oscillation: (1) free oscillation, (2) forced excited oscillation and (3) self-excited oscillation.
Free oscillation is defined as temporal fluctuations of the state variables of a system. Such temporal fluctuations can be defined as deviations from a mean value. Vibrations are present in many mechanical systems and occur always in feedback systems. The concept of free oscillation is misleading since nearly all physical systems are subject to attenuation. However, it depends on the size (and thus the time). Exceptions could be, for example, orbit oscillations of planets (macroscopic) or oscillations of electrons (microscopic). The two systems mentioned are also subjected to a type of damping, since both systems cannot remain stable indefinitely, but for an extremely long time.
A forced excited spring mass system might be a mechanically forced oscillator. Such systems of translational motions are discussed in Sections 2 and 3. Beside translatory oscillations, rotatory oscillations and resonance is of vital interest to design engineers of aircraft turbines, etc. Unbalanced rotating machine parts are sources of unwanted vibrations and might resonate when excited accordingly.
Self-excited oscillation, also called as self-oscillation, self-induced, maintained or autonomous oscillation is known in electronics as parasitic oscillation and in mechanical engineering literature as hunting. Such systems are discussed in Section 3.
Table 1 depicts relevant parameters for characterization motion in translational and rotational structures. The parameters for displacement, velocity and acceleration have been written as absolute values – knowing that depending on the application, they might be vectors, depending on the chosen frame of reference. In the most general case, they form a four vector. The force is written as mass times acceleration (Newton’s second law) and therefore force is also a vector. That brings us to Newton’s first law, which states that an object that is at rest will stay at rest unless a force acts upon it or inversely an object will not change its velocity unless a force acts upon it. For completeness, also Newton’s third law shall be given: Actio et Reactio – all forces between two objects exist in equal magnitude and opposite direction. A treaty to Newton’s laws of dynamics can be found, for example in chapter 9 of volume I .
|Symbol||Description||SI Unit||Symbol||Description||SI Unit|
|Kinetic energy||Nm||Kinetic Energy||Nm|
|Potential energy||Nm||Potential energy||Nm|
D'Alembert’s principle is a statement of the fundamental classical laws of motion. It is the dynamic analogue to the principle of virtual work for applied forces in a static system and in fact is more general than Hamilton’s principle, avoiding restriction to holonomic systems1. If the negative terms in accelerations are recognized as inertial forces, the statement of d'Alembert's principle becomes “the total virtual work of the impressed forces plus the inertial forces vanishes for reversible displacements”. The principle does not apply for irreversible displacements, such as sliding friction, and more general specification of the irreversibility is required. A derivation of the Lagrangian equation of motion is well explained in Chapter 1 of  or . In (1), the non-conservative energy term defined as the Lagrangian (
In (2) the Lagrangian equation is given with generalized coordinates
The sum of all kinetic energies
Looking at translational (classic) mechanical springs, the displacement dependent force
The elastic energy
The elastic energy
The elastic rotational energy can be expressed as follows (6):
In Figure 1, sketches on the top show translational springs (from left to right linear, nonlinear and nonlinear unsymmetrical) and sketches on the bottom depict rotational springs (from left to right linear, nonlinear and nonlinear unsymmetrical). The origin is depicted with an O and the spring displacement is depicted as
The rotational magnetic spring systems have a ferromagnetic stator fixed to the reference frame and a rotating hollow shaft carrying two permanent magnets, in this scenario also made of ferromagnetic material. The spring is drawn in the unstable equilibrium position. Exemplarily spring characteristics for such translational and rotational spring systems are depicted in Figure 2.
Note that integrating the force or torque response will add up to zero for rotational and translational spring systems. The given exemplarily spring characteristics of drawn spring systems in Figure 1 are shown in Figure 2. On the left-hand side, exemplarily translational spring characteristics are shown and on the right-hand side rotational spring characteristics are depicted. The linear case where the force and torque are proportional to the displacement is shown in blue. In red, symmetric nonlinear – here for translational and rotational systems a permanent magnet system is shown, but also mechanical spring systems could be envisaged. The curves of the nonlinear asymmetric cases are depicted in orange. In the appendix A.1, more simulations have been depicted for translational symmetric spring systems using ring magnets.
Equivalence of electrical and mechanical systems are shown in Table 2. On the left-hand side, a mechanical system with only one degree of freedom in
Applying the Lagrangian formalism (1) and (2) to these SDoF systems, will lead to the resulting DE’s as shown in (9) and (10). Note that the sign of the (viscous) damping must be introduced always with a negative sign using the generalized coordinates – as its velocity is always opposing the system velocity. In the electrical circuit, having the flowing charge velocity
2. Linear resonance systems
2.1. Linear single degree of freedom systems
In this section, a linear basepoint excited single degree of freedom systems is discussed. The lumped parameter model for the examined system (Figure 3) consists of a linear oscillator with mass
The kinetic energy (11) of this system and the potential energy (12) form the non-dissipative energy of this system. The dissipative force
Applying the Lagrangian formalism (1) and (2), we deal with SDoF system, will lead to the resulting DE shown in (15). Note that the sign of the viscous damping must be introduced with a negative sign using the generalized coordinates. The driving force
Introducing dimensionless notation, by using a dimensionless time
2.2. Linear two degree of freedom systems (2DoF systems)
In this section, a linear basepoint excited two degree of freedom systems is discussed. The lumped parameter model for the examined system (Figure 5) consists of two linear oscillators with mass
Similar to dimensionless parameters of (17), (18), the dimensionless time
The frequency response of this coupled oscillator system can again be obtained using the Laplace transformation introduced in (21). The system in the frequency domain is shown in (36) and (37) using the same steps as shown in the SDoF system (22)–(25).
As (36) and (37) represent two algebraic equations,
Figure 6 depicts the relative oscillation response in the frequency (left) and time domain (right) of the derived 2DoF system. The frequency response is given as a dimensionless ratio Ω, see also (34). The dimensionless simulation parameters have been set exemplarily to
The time-domain response from this coupled DE system with lumped parameter model Figure 5 and (32) and (33) is shown in Figure 7. On the left-hand side, the ^dimensionless basepoint acceleration signal is given and its dimensionless response signals of first (blue) and second (red) DoF, simulating 50 periods and starting with settled initial conditions (amplitude of ). The main simulation parameters are shown in the heading, a variant of this setup using nonlinear springs is given in the appendix A.3.
3. Nonlinear resonance systems
In the introduction, Section 1, we distinguished three cases of vibration. The class forced excitation will be further investigated in this section. In Figure 8 five systems are depicted that can potentially exhibit parametric resonance effects. The term parametric means that of cases where the external excitation appears as a time varying modification of a system parameter. A “normal” forced excitation system whether linear or nonlinear, will respond to the excitation with or without resonance using the energy fed into it and no time varying modification of a system parameter might excite additionally the system.
The five depicted systems in Figure 8 might show an exponential amplitude growth when excited externally in presence of a system damping factor. In the two electrical systems on the right-hand side, any of the three components R (here not drawn), L or C that is parametrically excited will respond with an exponential amplitude growth, if the mathematical physical system model has at least one degree of freedom of the Mathieu DE (40) or the Hill DE (41).
It is most interestingly that any system parameter including also damping factors with time varying influence of a system parameter will result in an exponential growth of the response amplitude. To give a concrete example of this behavior, we consider here the example from Section 3.2 and inspect the resulting (dimensioned) DE system with (62) as primary system and (63) as secondary system of such a behavior.
The primary system has no such configuration, but the secondary system (63) is of Mathieu type. To simplify the treated system, we use instead of the basepoint excitation
Rearranging the terms and setting sin
For generating parametric resonances, the (natural) system frequency needs to be coupled with the excitation frequency ω. Using the same nomenclature as in (64), we define the pendulum system frequency . For generating parametric resonances for which the angle
Note that the response signal in orange on the left-hand side of Figure 10 is using an approximated linear displacement function sin(
3.1. Nonlinear single degree of freedom systems
Similar to the case in Section 2.1, also a SDoF system will be discussed, but this time a linear and a nonlinear spring will be present. The nonlinearity of this spring shall have the form shown in (3) having a nonlinear exponent
The elastic energy of this nonlinear spring system with
Applying the Lagrangian formalism (1) and (2), we deal again with a SDoF system, will lead to the resulting DE shown in (50), similar to the result derived in Section 2.1 – but here we have now introduced a nonlinear spring system.
Introducing dimensionless notation, by using a dimensionless time
DE (53) is nonlinear, as we have also a path dependent function to the power of 3 and a dimensionless factor
The depicted Figure 11 shows simulation results of DE (53). The time domain behavior (top left) and its dimensionless phase space behavior (top right) is shown with simulating 50 periods with settled initial conditions (amplitude of . The top row depicts one simulation point in the bottom row, where a frequency sweep has been done, sweeping the basepoint excitation from
3.2. Nonlinear two degree of freedom systems
Let us consider the lumped parameter model in Figure 12. A pendulum with a stiff rod of length
Governing equations are derived using again the Lagrange formalism. Considering the frame of reference at the origin shown in Figure 12 and defining in Cartesian coordinates first the two degrees of freedom vector
The Lagrange energy function
Applying the Lagrange formalism (2) for both degrees
The parameters for non dimensionalization are given in (64), (65). Note that the reference system frequency
Figure 13 depicts left the excitation (magenta) and the time response signals of
A treaty of such a system, a kinetic energy harvesting device, with additionally a nonlinear spring system on the primary and an electromagnetic harvester on the secondary system is given in Ref. . Note that the derivation of the system equations there have been made without the Lagrangian formalism and the found system equations are equivalent.
In the introduction, we showed the equivalence of rotary and translatory mechanical systems as well as the equivalence of mechanical and electrical resonance systems. Also, a brief introduction to the Lagrangian formalism is given. In preparation to nonlinear resonance systems, also rotational and translational springs are discussed. Three classes of spring systems have been identified: linear springs nonlinear symmetric springs and nonlinear asymmetric springs. Throughout the chapter further readings are proposed.
In Section 2, linear resonance systems with one and two degrees of freedom have been investigated using basepoint excited systems. Using the Laplace Transformation is most useful to analyze any linear resonance system with a periodic excitation.
Section 3 deals with nonlinear resonance systems. When in such a dynamical system one of the resulting DE’s is of Mathieu or Hill type, the response amplitude of such a system might grow exponentially. This is exemplary demonstrated in Section 3.2 identifying the system differential equations of a basepoint excited two degree of freedom system. Some dynamic properties of such a system is demonstrated.
A.1. Nonlinear symmetric spring systems
Using instead of disk magnets ring magnets, strong nonlinearities can be generated. The following series in Figure A1 depicts a few simulation cases. Some of shown simulation cases have been validated and proven experimentally.
A.2. Variant of linear 2DoF system
Instead of using linear springs, magnetic nonlinear springs can be used (see also a selection of such spring characteristics in A.1). Using nonlinear springs and making the system nonlinear (instead of having only a linear spring term, we have for each spring also a term of the form
It is interesting, that the relative motion of the 2. DoF is responding with resonance between 19…25.5 Hz.The first degree of freedom has a nonlinear spring hardening behavior, reaching
A.3. Variant of nonlinear SDoF system
A created tool by the author in Matlab/Simulink has been used to simulate many basepoint excited SDoF or nDoF systems with rotational or translational or mixed structures.
It allows to simulate such systems with constant amplitude or constant acceleration, can handle hard or soft-impact of the oscillating proof mass(es). In addition, one sided spring characteristics can be simulated, see also Figure A3 – a feature that is especially interesting in relation with magnetic springs. Main disadvantage of such one-sided bound springs is the fact, that they need to be installed upright. The behavior of such a one-sided magnetic spring is depicted in Figure A3. It has a frequency response similar to a softening spring. The maximal amplitude of 3.65 mm occurs at 17.5 Hz (the two-sided classical hardening magnetic spring reaches an amplitude maximum of 4.3 mm at 27 Hz). Such a spring system could also be analytically described, by introducing for example continuous piecewise functions.
- A holonomic constraint depends only on the coordinates and time and does not depend on velocities.