Open access peer-reviewed chapter
By Nguyen Van Khang and Nguyen Phong Dien
Submitted: February 17th 2012Reviewed: June 29th 2012Published: October 2nd 2012
DOI: 10.5772/51157
Downloaded: 1463
Transmission mechanisms are frequently used in machines for power transmission, variation of speed and/or working direction and conversion of rotary motion into reciprocating motion. At high speeds, the vibration of mechanisms causes wear, noise and transmission errors. The vibration problem of transmission mechanisms has been investigated for a long time, both theoretically and experimentally. In dynamic modelling, a transmission mechanism is usually modelled as a multibody system. The differential equations of motion of a multibody system that undergo large displacements and rotations are fully nonlinear in n generalized coordinates in vector of variable q [1–4].
It is very difficult or impossible to find the solution of Eq. (1) with the analytical way. Nevertheless, the numerical methods are efficient to solve the problem [5-9].
Besides, many technical systems work mostly on the proximity of an equilibrium position or, especially, in the neighbourhood of a desired motion which is usually called “programmed motion”, “desired motion”, “fundamental motion”, “input–output motion” and etc. according to specific problems. In this chapter, the term “desired fundamental motion“ is used for this object. The desired fundamental motion of a robotic system, for instance, is usually described through state variables determined by prescribed motions of the end-effector. For a mechanical transmission system, the desired fundamental motion can be the motion of working components of the system, in which the driver output rotates uniformly and all components are assumed to be rigid. It is very convenient to linearize the equations of motion about this configuration to take advantage of the linear analysis tools [10-18]. In other words, linearization makes it possible to use tools for studying linear systems to analyze the behavior of multibody systems in the vicinity of a desired fundamental motion. For this reason, the linearization of the equations of motion is most useful in the study of control [12-13], machinery vibrations [14-19] and the stability of motion [20-21]. Mathematically, the linearized equations of motion of a multibody system form usually a set of linear differential equations with time-varying coefficients. Considering steady-state motions of the multibody system only, one obtains a set of linear differential equations having time-periodic coefficients.
Note that Eq. (2) can be expressed in the compact form as
where we use the state variable x
and the matrix of coefficients P(t), vector f(t) are defined by
where I denotes the
In the steady state of a machine, the working components perform stationary motions [14-18], matrices
Since periodic vibrations are a commonly observed phenomenon of transmission mechanisms in the steady-state motion, a number of methods and algorithms were developed to find a T-periodic solution of the system described by Eq. (2). A common approach is by imposing an arbitrary set of initial conditions, and solving Eq. (2) in time using numerical methods until the transient term of the solution vanishes and only the periodic steady-state solution remains [14,22]. Besides, the periodic solution can be found directly by other specialized techniques such as the harmonic balance method, the method of conventional oscillator, the WKB method [14-16, 23, 24].
Following the above introduction, an overview of the numerical calculation of dynamic stability conditions of linear dynamic systems with time-periodic coefficients is presented in Section 2. Sections 3 presents numerical procedures based on Runge-Kutta method and Newmark method to find periodic solutions of linear systems with time-periodic coefficients. In Section 4, the proposed approach is demonstrated and validated by dynamic models of transmission mechanisms and measurements on real objects. The improvement in the computational efficiency of Newmark method comparing with Runge-Kutta method for linear systems is also discussed.
We shall consider a system of homogeneous differential equations
where
is called the monodromy matrix of Eq. (6) [20]. The characteristic equation of Eq. (6) can then be written in the form
Expansion of Eq. (9) yields a n-order algebraic equation
where unknowns
When the Floquet multipliers or Floquet exponents are known, the stability conditions of solutions of the system of linear differential equations with periodic coefficients can be easily determined according to the Floquet theorem [17–20]. The concept of stability according to Floquet multipliers can be expressed as follows.
If
If
If
Now we consider only the periodic vibration of a dynamic system which is governed by a set of linear differential equations with periodic coefficients. As already mentioned in the previous section, these differential equations can be expressed in the compact matrix form
where x is the vector of state variables, matrix
As well known from the theory of differential equations, if Eq. (13) has only non-periodic solutions except the trivial solution, then Eq. (12) has an unique T-periodic solution. This periodic solution can be obtained by choosing the appropriate initial condition for the vector of variables x and then implementing numerical integration of Eq. (12) within interval
The interval
where
Substituting Eq. (16) into Eq. (15), we obtain
where matrix
and vector
Expansion of Eq. (17) for
where
The solution of Eq. (21) gives us the initial value for the periodic solution of Eq. (12). Finally, the periodic solution of Eq. (12) with the corresponding initial value can be calculated using the computational scheme according to Eq. (15).
The procedure presented below for finding the T-periodic solution of Eq. (2) is based on the Newmark direct integration method. Firstly, the interval
Based on the single-step integration method proposed by Newmark, we obtain the following approximation formulas [6-7]
Constants
From Eq. (2) we have the following iterative computational scheme at time
where
In the next step, substitution of Eqs. (23) and (24) into Eq. (25) yields
The use of Eqs. (23) and (24) leads to the prediction formulas for velocities and displacements at time
Eq. (27) can be expressed in the matrix form as
with
where 0 represents the
where matrices
By substituting relationships (28) into (30) we find
From Eqs. (23), (24) and (27) we get the following matrix relationship
where matrix T is expressed in the block matrix form as
The combination of Eqs. (28), (33) and (34) yields a new computational scheme for determining the solution of Eq. (2) at the time
In this equation, the iterative computation is eliminated by introducing the direct solution for each time step. Note that T and D are matrices of constants.
By setting
Eq. (36) can then be rewritten in the following form
Expansion of Eq. (38) for
where
Using the condition of periodicity according to Eq. (22), the last equation of Eq. (39) yields a set of the linear algebraic equations
The solution of Eq. (40) gives us the initial value for the periodic solution of Eq. (2). Finally, the periodic solution of Eq. (2) with the obtained initial value can be calculated without difficulties using the computational scheme in Eq. (36).
Based on the proposed numerical procedures in this section, a computer program with MATLAB to calculate periodic vibrations of transmission mechanisms has been developed and tested by the following application examples.
Cam mechanisms are frequently used in mechanical transmission systems to convert rotary motion into reciprocating motion (Figure 1). At high speed, the vibration of cam mechanisms causes transmission errors, cam surface fatigue, wear and noise. Because of that, the vibration problem of cam mechanisms has been investigated for a long time, both theoretically and experimentally.
A cam mechanism.
The dynamic model of this system is schematically shown in Figure 2. This kind of model was also considered in a number of studies, e.g. [25-26]. The mechanical system of the elastic cam shaft, the cam with an elastic follower can be considered as rigid bodies connected by massless spring-damping elements with time-invariant stiffness
Dynamic model of the cam mechanism.
The kinetic energy, the potential energy and the dissipative function of the considered system can be expressed in the following form
The virtual work done by all non-conservative forces is
Using the generalized coordinates
Substitution of Eq. (45) into Eqs. (41-44) yields
where the prime represents the derivative with respect to the generalized coordinate
Substitution of Eqs. (46)-(48) and (50) into the Lagrange equation of the second type yields the differential equations of motion of the system in terms of the generalized coordinates
When the angular velocity
one leads to the following relation
where
where we used the notations
Since the system performs small vibrations, i.e. there are only small vibrating amplitudes
In most cases, the force
where
We consider now the function
| Parameters | Units | Values |
| (kg) | 28 | |
| (kg) | 50 | |
| (kgm2) | 0.12 | |
| (Nm/rad) | ||
| ( N/m) | ||
| ( N/m) | ||
| (Nms/rad) | 18.5 | |
| (Ns/m) | 1400 | |
| (Ns/m) | 1200 |
Calculation parameters.
The functions
| a k ( m) | Case 1 | Case 2 |
| a 1 | 0.22165 | 0.22206 |
| a 2 | 0 | 0 |
| a 3 | 0.05560 | 0.08539 |
| a 4 | 0 | 0 |
| a 5 | - 0.01706 | 0.00518 |
| a 6 | 0 | 0 |
| a 7 | 0 | - 0.00373 |
| a 8 | 0 | 0 |
| a 9 | 0 | 0.00345 |
| a 10 | 0 | 0 |
| a 11 | 0 | - 0.00182 |
| a 12 | 0 | 0 |
Fourier coefficients
Dynamic transmission errors q 3 with nim=100(rpm) for Case 1 (left) and Case 2 (right).
The rotating speed of the driver input
Frequency spectrum of q 3 with n i n = 100 ( r p m ) for Case 1 (left) and Case 2 (right).
Figures 5 and 6 show the calculating results with rotating speed
Dynamic transmission errors q 3 with n i n = 600 ( r p m ) for Case 1.
Dynamic transmission errors q 3 with n i n = 600 ( r p m ) for Case 2.
The calculation of the periodic solution of Eq. (64) was implemented by a self-written computer program in MATLAB environment, and a Dell Notebook equipped with CPU Intel® Core 2 Duo T6600 at 2.2 GHz and 3 GB memory. The calculating results obtained by the numerical procedures are identical, but the computation time with Newmark method is greatly reduced in comparison with Runge-Kutta method as shown in Figure 7, especially in the cases of large number of time steps.
Comparison of the computation time for the model of cam mechanism.
Dynamic modeling of gear vibrations offers a better understanding of the vibration generation mechanisms as well as the dynamic behavior of the gear transmission in the presence of gear tooth damage. Since the main source of vibration in a geared transmission system is usually the meshing action of the gears, vibration models of the gear-pair in mesh have been developed, taking into consideration the most important dynamic factors such as effects of friction forces at the meshing interface, gear backlash, the time-varying mesh stiffness and the excitation from gear transmission errors [31-33].
Dynamic model of the gear-pair system with faulted meshing.
From experimental works, it is well known that the most important components in gear vibration spectra are the tooth-meshing frequency and its harmonics, together with sideband structures due to the modulation effect. The increment in the number and amplitude of sidebands may indicate a gear fault condition, and the spacing of the sidebands is related to their source [27], [30]. However, according to our knowledge, there are in the literature only a few of theoretical studies concerning the effect of sidebands in gear vibration spectrum and the calculating results are usually not in agreement with the measurements. Therefore, the main objective of the following investigation is to unravel modulation effects which are responsible for generating such sidebands.
Figure 8 shows a relative simple dynamic model of a pair of helical gears. This kind of the model is also considered in references [24, 28, 32, 33]. The gear mesh is modeled as a pair of rigid disks connected by a spring-damper set along the line of contact.
The model takes into account influences of the static transmission error which is simulated by a displacement excitation e(t) at the mesh. This transmissions error arises from several sources, such as tooth deflection under load, non-uniform tooth spacing, tooth profile errors caused by machining errors as well as pitting, scuffing of teeth flanks. The mesh stiffness
where
Eqs. (66) and (67) yield a single differential equation in the following form
where
Note that the rigid-body rotation from the original mathematical model in Eqs. (66) and (67) is eliminated by introducing the new coordinate q(t) in Eq. (69). Variable q(t) is called the dynamic transmission error of the gear-pair system [32]. Upon assuming that when
Eq. (69) can then be rewritten in the form
where
In steady state motion of the gear system, the mesh stiffness k z (t) can be approximately represented by a truncated Fourier series [33]
where
In general, the error components are no identical for each gear tooth and will produce displacement excitation that is periodic with the gear rotation (i.e. repeated each time the tooth is in contact). The excitation function e(t) can then be expressed in a Fourier series with the fundamental frequency corresponding to the rotation speed of the faulted gear. When the errors are situated at the teeth of the pinion, e(t) may be taken in the form
| Parameters | Pinion | Wheel |
| Gear type | helical, standard involute | |
| Material | steel | |
| Module (mm) | 4.50 | |
| Pressure angle (o) | 20.00 | |
| Helical angle (o) | 14.56 | |
| Number of teeth z | 14 | 39 |
| face width (mm) | 67.00 | 45.00 |
| base circle radius (mm) | 30.46 | 84.86 |
Parameters of the test gears.
Therefore, the vibration equation of gear-pair system according to Eq. (72) is a differential equation with the periodic coefficients.
According to the experimental setup which will be described later, the model parameters include
| n | k n (N/m) | γ n (radian) |
| 0 | 8.1846108 | |
| 1 | 3.2267107 | 2.5581 |
| 2 | 1.3516107 | -1.4421 |
| 3 | 8.1510106 | -2.2588 |
| 4 | 3.5280106 | 0.9367 |
| 5 | 4.0280106 | -0.8696 |
| 6 | 9.7100105 | -2.0950 |
| 7 | 1.4245106 | 0.9309 |
| 8 | 1.5505106 | 0.2584 |
| 9 | 4.6450105 | -1.2510 |
| 10 | 1.4158106 | 2.1636 |
Fourier coefficients and phase angles of the mesh stiffness.
| i | Case 1 | Case 2 | ||
| e i (mm) | α i (rad) | e i (mm) | α i (rad) | |
| 1 | 0.0015 | -0.049 | 0.010 | 1.0470 |
| 2 | 0.0035 | -1.7661 | 0.003 | -1.4521 |
| 3 | 0.0027 | -0.7286 | 0.0018 | 0.5233 |
| 4 | 0.0011 | -0.5763 | 0.0011 | 1.4570 |
| 5 | 0.0005 | -0.7810 | 0.0009 | -0.8622 |
| 6 | 0.0013 | 1.8172 | 0.0003 | 1.1966 |
Fourier coefficients and phase angles of excitation function
Using the obtained periodic solutions of Eq. (72), the calculated dynamic transmission errors are shown in Figures 9 and 10 corresponding to different excitation functions
. Modelling result: dynamic transmission error q(t).
Modelling result: frequency spectrum of dq/dt corresponding to (a) excitation function e(t) of Case 1 and b) excitation function e(t) with larger coefficients (Case 2).
The experiment was done at an ordinary back-to-back test rig (Figure 11). The major parameters of the test gear-pair are given in Table 3. The load torque was provided by a hydraulic rotary torque actuator which remains the external torque constant for any motor speed. The test gearbox operates at a nominal pinion speed of 1800 rpm. (30 Hz), thus the meshing frequency f z is 420 Hz. A Laser Doppler Vibrometer was used for measuring oscillating parts of the angular speed of the gear shafts (i.e. oscillating part of
Gearbox test rig.
Experimental result: frequency spectrum of dq/dt.
Figure 12 shows a frequency spectrum of the first derivative of the dynamic transmission error
Calculating result: frequency spectrum of dq/dt.
Experimental result: zoomed frequency spectrum of dq/dt from Figure 12.
The calculations required a large number of time steps to ensure that the frequency resolution in vibration spectra is fine enough. In comparison with the numerical procedure based on Runge-Kutta method, the computation time by the Newmark-based numerical procedure is greatly reduced for large number of time steps as shown in Figure 15, for that the same computer was used as in the previous example.
Comparison of the computation time for the gear-pair model.
The most common forging equipment is the mechanical forging press. Mechanical presses function by using a transport manipulator with a cam mechanism to produce a preset at a certain location in the stroke. The kinematic schema of such mechanical adjustment unit is depicted in Figure 16.
Kinematic schema of the transport manipulator of a forging press: 1- the first gearbox, 2- driving shaft, 3- the second gearbox, 4- cam mechanism, 5- operating mechanism (hammer).
The dynamic model of this system shown in Figure 17 is used to investigate periodic vibrations which are a commonly observed phenomenon in mechanical adjustment unit during the steady-state motion [18, 23]. The system of the driver shaft, the flexible transmission mechanism and the hammer can be considered as rigid bodies connected by spring-damping elements with time-invariant stiffness
Dynamic model of the transport manipulator.
When the angular velocity
one leads to the following relation
where
By the analogous way as in Section 3.1, we obtain the linear differential equations of vibration for the system in the compact matrix form as
where
In steady state motion of the cam mechanism, function
The functions
The following parameters are used for numerical calculations: Rotating speed of the driver input
Calculating result of q 2 for case 1, (a) time curve, (b) frequency spectrum.
Calculating result of q 2 for case 2, (a) time curve, (b) frequency spectrum.
The Fourier coefficients
To verify the dynamic stable condition of the vibration system, the maximum of absolute value
Calculating results of periodic vibrations of the mechanical adjustment unit, i.e. periodic solutions of Eq. (77), are shown in Figures 18-19 for two cases of the cam profile. Comparing both time curves, the influence of cam profiles on the vibration level of the hammer can be recognized. In addition, the frequency spectrums show harmonic components of the rotating frequency at Ω, 3Ω, 5Ω. These spectrums indicate that the considered system performs stationary periodic vibrations only.
Dynamic moment acting on the driving shaft of the mechanical adjustment unit
To verify the calculating results using the numerical methods, the dynamic load moment of the mechanical adjustment unit was measured on the driving shaft (see also Figure 16). A typical record of the measured moment is plotted in Figure 20, together with the curves calculated from the dynamic model by using the WKB-method [18, 34], the kinesto-static calculation and the proposed numerical procedures based on Newmark method and Runge-Kutta method. Comparing the curves displayed in this figure, it can be observed that the calculating result using the numerical methods is more closely in agreement with the experimental result than the results obtained by the WKB-method and the kinesto-static calculation.
The calculation of dynamic stable conditions and periodic vibrations of elastic mechanisms and machines is an important problem in mechanical engineering. This chapter deals with the problem of dynamic modelling and parametric vibration of transmission mechanisms with elastic components governed by linearized differential equations having time-varying coefficients.
Numerical procedures based on Runge-Kutta method and Newmark integration method are proposed and applied to find periodic solutions of linear differential equations with time-periodic coefficients. The periodic solutions can be obtained by Newmark based procedure directly and more conveniently than the Runge-Kutta method. It is verified that the computation time with the Newmark based procedure reduced by about 60%-65% compared to the procedure using the fourth-order Runge-Kutta method (see also Figures 7 and 15). Note that this conclusion is only true for linear systems.
The numerical methods and algorithms are demonstrated and tested by three dynamic models of elastic transmission mechanisms. In the last two examples, a good agreement is obtained between the model result and the experimental result. It is believed that the proposed approaches can be successfully applied to more complicated systems. In addition, the proposed numerical procedures can be used to estimate approximate initial values for the shooting method to find the periodic solutions of nonlinear vibration equations.
This study was completed with the financial support by the Vietnam National Foundation for Science and Technology Development (NAFOSTED).
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