Open access peer-reviewed chapter

# Parametric Vibration Analysis of Transmission Mechanisms Using Numerical Methods

By Nguyen Van Khang and Nguyen Phong Dien

Submitted: February 17th 2012Reviewed: June 29th 2012Published: October 2nd 2012

DOI: 10.5772/51157

## 1. Introduction

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 ngeneralized coordinates in vector of variable q [14].

M(q,t)q¨+k(q˙,q,t)=h(q˙,q,t)E1

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.

M(t)q¨(t)+C(t)q˙(t)+K(t)q(t)=d(t)E2

Note that Eq. (2) can be expressed in the compact form as

x˙=P(t)x+f(t)E3

where we use the state variable x

x=[qq˙],x˙=[q˙q¨]E4

and the matrix of coefficients P(t), vector f(t) are defined by

P(t)=[0IM1KM1C],f(t)=[0M1d],E5

whereIdenotes the n×nidentity matrix.

In the steady state of a machine, the working components perform stationary motions [14-18], matrices M(t),C(t),K(t)and vector d(t)in Eq. (2) are time-periodic with the least period T. Hence, Eq. (2) represents a parametrically excited system. For calculating the steady-state periodic vibrations of systems described by differential equations (1) or (2) the harmonic balance method, the shooting method and the finite difference method are usually used [8,11,14]. In addition, the numerical integration methods as Newmark method and Runge-Kutta method can also be applied to calculate the periodic vibration of parametric vibration systems governed by Eq. (2) [5-9].

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.

## 2. Numerical calculation of dynamic stability conditions of linear dynamic systems with time-periodic coefficients: An overview

We shall consider a system of homogeneous differential equations

x˙=P(t)xE6

where P(t)is a continuous T-periodic n×nmatrix. According to Floquet theory [17, 18, 20, 21], the characteristic equation of Eq. (6) is independent of the chosen fundamental set of solutions. Therefore, the characteristic equation can be formulated by the following way. Firstly, we specify a set of ninitial conditions xi(0)fori=1,...,n, their elements

xi(s)(0)={1whens=i0otherwhileE7
and [x1(0),x2(0),...,xn(0)]=I.By implementing numerical integration of Eq. (6) within interval [0,T]for ngiven initial conditions respectively, we obtain nvectorsxi(T),i=1,...,n. The matrix Φ(t)defined by
Φ(T)=[x1(T),x2(T),...,xn(T)]E8

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

ρn+a1ρn1+a2ρn2+....+an1ρ+an=0E10

where unknowns ρk(k=1,...,n),called Floquet multipliers, can be determined from Eq. (10). Floquet exponents are given by

λk=1Tlnρk,(k=1,...,n)E11

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 [1720]. The concept of stability according to Floquet multipliers can be expressed as follows.

1. If |ρk|1,the trivial solution x=0of Eq. (6) will be asymptotically stable. Conversely, the solution x=0of Eq. (6) becomes unstable if at least one Floquet multiplier has modulus being larger than 1.

2. If |ρk|1and Floquet multipliers with modulus 1 are single roots of the characteristic equation, the solution x=0of Eq. (6) is stable.

3. If |ρk|1and Floquet multipliers with modulus 1 are multiple roots of the characteristic equation, and the algebraic multiplicity is equal to their geometric multiplicity, then the solution x=0of Eq. (6) is also stable.

## 3. Numerical procedures for calculating periodic solutions of linear dynamic systems with time-periodic coefficients

### 3.1. Numerical procedure based on Runge-Kutta method

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

x˙=P(t)x+f(t)E12

where xis the vector of state variables, matrix P(t)and vector f(t)are periodic in time with period T. The system of homogeneous differential equations corresponding to Eq. (12) is

x˙=P(t)xE13

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[0,T]. An algorithm is developed to find the initial value for the periodic solution [18, 19]. Firstly, the T-periodic solution must satisfy the following condition

x(0)=x(T)E14

The interval [0,T]is now divided into mequal subintervals with the step-size h=titi1=T/m.At the discrete times tiand ti+1,xi=x(ti)and xi+1=x(ti+1)represent the states of the system, respectively. Using the fourth-order Runge-Kutta method, we get a numerical solution [5]

xi=xi1+16[k1(i1)+2k2(i1)+2k3(i1)+k4(i1)]E15

where

k1(i1)=h[P(ti1)xi1+f(ti1)],k2(i1)=h[P(ti1+h2)(xi1+12k1(i1))+f(ti1+h2)],k3(i1)=h[P(ti1+h2)(xi1+12k2(i1))+f(ti1+h2)],k4(i1)=h[P(ti)(xi1+k3(i1))+f(ti)].E16

Substituting Eq. (16) into Eq. (15), we obtain

xi=Ai1xi1+bi1E17

where matrix Ai1is given by

Ai1=I+16{h[P(ti1)+4P(ti1+h2)+P(ti)]+h2[P(ti1+h2)P(ti1)+P2(ti1+h2)+12P(ti)P(ti1+h2)]+h32[P2(ti1+h2)P(ti1)+12P(ti)P2(ti1+h2)]+h44P(ti)P2(ti1+h2)P(ti1)}(i=1,...,m),E18

and vector bi1takes the form

bi1=16{h[f(ti1)+4f(ti1+h2)+f(ti)]+h2[P(ti1+h2)f(ti1)+P(ti1+h2)f(ti1+h2)+12P(ti)f(ti1+h2)]+h32[P2(ti1+h2)f(ti1)+P(ti)P(ti1+h2)f(ti1+h2)]+h44P(ti)P2(ti1+h2)f(ti1)}.E19

Expansion of Eq. (17) for i=1tomyields

x1=A0x0+c1x2=A1A0x0+c2................................xm=(i=m10Ai)x0+cmE20

where c0=0,c1=A0c0+b0,c2=A1c1+b1,..., cm=Am1cm1+bm1.Using the boundary condition according to Eq. (14), the last equation of Eq. (20) yields a set of the linear algebraic equations

(Ii=m10Ai)x0=cm.E21

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).

### 3.2. Numerical procedure based on Newmark integration method

The procedure presented below for finding the T-periodic solution of Eq. (2) is based on the Newmark direct integration method. Firstly, the interval [0,T]is also divided into mequal subintervals with the step-size h=titi1=T/m.We use notations qi=q(ti)and qi+1=q(ti+1)to represent the solution of Eq. (2) at discrete times tiand ti+1respectively. The T-periodic solution must satisfy the following conditions

q(0)=q(T),q˙(0)=q˙(T),q¨(0)=q¨(T).E22

Based on the single-step integration method proposed by Newmark, we obtain the following approximation formulas [6-7]

qi+1=qi+hq˙i+h2(12β)q¨i+βh2q¨i+1,E23
q˙i+1=q˙i+(1γ)hq¨i+γhq¨i+1,E24

Constantsβ,γare parameters associated with the quadrature scheme. Choosing γ=1/4and β=1/6leads to linear interpolation of accelerations in the time interval [ti,ti+1]. In the same way, choosing γ=1/2,β=1/4corresponds to considering the acceleration average value over the time interval [6, 7].

From Eq. (2) we have the following iterative computational scheme at timeti+1

Mi+1q¨i+1+Ci+1q˙i+1+Ki+1qi+1=di+1,E25

where Mi+1=M(ti+1),Ci+1=C(ti+1),Ki+1=K(ti+1)and di+1=d(ti+1).

In the next step, substitution of Eqs. (23) and (24) into Eq. (25) yields

(Mi+1+γhCi+1+βh2Ki+1)q¨i+1=di+1Ci+1[q˙i+(1γ)hq¨i]Ki+1[qi+hq˙i+h2(12β)q¨i].E26

The use of Eqs. (23) and (24) leads to the prediction formulas for velocities and displacements at time ti+1

qi+1*=qi+hq˙i+h2(12β)q¨i,q˙i+1*=q˙i+(1γ)hq¨i.E27

Eq. (27) can be expressed in the matrix form as

[qi+1*q˙i+1*]=D[qiq˙iq¨i]E28

with

D=[IhIh2(0.5β)I0I(1γ)hI]E29

where 0represents the n×nmatrix of zeros. Eq. (26) can then be rewritten in the matrix form as

q¨i+1=(Si+1)1di+1(Si+1)1Hi+1[qi+1*q˙i+1*],E30

where matrices Si+1and Hi+1are defined by

Si+1=Mi+1+γhCi+1+h2βKi+1,E31
Hi+1=[Ki+1Ci+1].E32

By substituting relationships (28) into (30) we find

q¨i+1=(Si+1)1di+1(Si+1)1Hi+1D[qiq˙iq¨i]E33

From Eqs. (23), (24) and (27) we get the following matrix relationship

[qi+1q˙i+1q¨i+1]=T[qi+1*q˙i+1*q¨i+1],E34

where matrix Tis expressed in the block matrix form as

T=[I0Iβh20IIγh00I]E35

The combination of Eqs. (28), (33) and (34) yields a new computational scheme for determining the solution of Eq. (2) at the time ti+1in the form

[qi+1q˙i+1q¨i+1]=T[D(Si+1)1Hi+1D][qiq˙iq¨i]+T[00(Si+1)1di+1]E36

In this equation, the iterative computation is eliminated by introducing the direct solution for each time step. Note that Tand Dare matrices of constants.

By setting

xi=[qiq˙iqi],Ai+1=T[D(Si+1)1Hi+1D],bi+1=T[00(Si+1)1di+1]E37

Eq. (36) can then be rewritten in the following form

xi=Aixi1+bi(i=1,2,...,m).E38

Expansion of Eq. (38) for i=1tomyields the same form as Eq. (20)

x1=A1x0+c1x2=A2A1x0+c2................................xm=(i=m1Ai)x0+cmE39

where c0=0,c1=A1c0+b1,c2=A2c1+b2,..., cm=Amcm1+bm.

Using the condition of periodicity according to Eq. (22), the last equation of Eq. (39) yields a set of the linear algebraic equations

(Ii=m1Ai)x0=cm.E40

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.

## 4. Application examples

### 4.1. Steady-state parametric vibration of an elastic cam mechanism

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.

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 kiand constant damping coefficients cifor i=1,2,3.Among them k1is the torsional stiffness of the cam shaft. Parameter k2is the equivalent stiffness due to the longitudinal stiffness of the follower, the contact stiffness between the cam and the roller, and the cam bearing stiffness. Parameter k3denotes the combined stiffness of the return spring and the support of the output link. The rotating components are modeled by two rotating disks with moments of inertia I0and I1.Let us introduce into our dynamic model the nonlinear transmission function U(φ1)of the cam mechanism as a function of the rotating angle φ1of the cam shaft, the driving torque from the motor M(t) and the external load F(t) applied on the system.

The kinetic energy, the potential energy and the dissipative function of the considered system can be expressed in the following form

T=12I0φ˙02+12I1φ˙12+12m2y˙22+12m3y˙32E41
Π=12k1(φ1φ0)2+12k2(y2y1)2+12k3(y3y2)2E42
Φ=12c1(φ˙1φ˙0)2+12c2(y˙2y˙1)2+12c3(y˙3y˙2)2E43

The virtual work done by all non-conservative forces is

δA=M(t)δφ0F(t)δy3E44

Using the generalized coordinates φ0,φ1,q2,q3,we obtain the following relations

y1=U(φ1),y2=y1+q2,y3=y2+q3E45

Substitution of Eq. (45) into Eqs. (41-44) yields

T=12I0φ˙02+12I1φ˙12+12m2(Uφ˙1+q˙2)2+12m3(Uφ˙1+q˙2+q˙3)2,E46
Π=12k1(φ1φ0)2+12k2q22+12k3q32,E47
Φ=12c1(φ˙1φ˙0)2+12c2q˙22+12c3q˙32,E48
δA=M(t)δφ0F(t)Uδφ1F(t)δq2F(t)δq3,E49

where the prime represents the derivative with respect to the generalized coordinate φ1.The generalized forces of all non-conservative forces are then derived from Eq. (49) as

Qφ0*=M(t),  Qφ1*=F(t)U,Qq2*=F(t),Qq3*=F(t).E50

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 φ0,φ1,q2,q3

I0φ¨0c1(φ˙1φ˙0)k1(φ1φ0)=M(t),E51
[I1+(m2+m3)U2]φ¨1+(m2+m3)Uq¨2+m3Uq¨3+(m2+m3)UUφ˙12+c1(φ˙1φ˙0)+k1(φ1φ0)=F(t)U,E52
(m2+m3)Uφ¨1+(m2+m3)q¨2+m3q¨3+(m2+m3)Uφ˙12+c2q˙2+k2q2=F(t),E53
m3Uφ¨1+m3q¨2+m3q¨3+m3Uφ˙12+c3q˙3+k3q3=F(t).E54

When the angular velocity Ωof the driver input is assumed to be constant in the steady state

φ0=Ωt,E55

one leads to the following relation

φ1=Ωt+q1,E56

where q1is the difference between rotating angles φ0and φ1due to the presence of the spring element k1and the damping element c1.Assuming that φ1varies little from its mean value during the steady-state motion, the transmission function y1=U(φ1)depends essentially on the input angle φ0=Ωt.Using the Taylor series expansion around Ωt,we get

U(φ1)=U(Ωt+q1)=U¯+U¯q1+12U¯q12+,E57
U(φ1)=U(Ωt+q1)=U¯+U¯q1+12U¯q12+,E58
U(φ1)=U(Ωt+q1)=U¯+U¯q1+12U¯(4)q12+.E59

where we used the notations

U¯=U(Ωt),U¯=U(Ωt),U¯=U(Ωt),U¯=U(Ωt).E60

Since the system performs small vibrations, i.e. there are only small vibrating amplitudes q1,q2and q3,substituting Eqs. (57)-(59) into Eqs. (52)-(54) and neglecting nonlinear terms, we obtain the linear differential equations of vibration for the system

(I1+(m2+m3)U¯2)q¨1+(m2+m3)U¯q¨2+m3U¯q¨3+[c1+2(m2+m3)ΩU¯U¯]q˙1+[k1+F(t)U¯+(m2+m3)Ω2(U¯U¯+U¯2)]q1=F(t)U¯(m2+m3)Ω2U¯U¯,E61
(m2+m3)U¯q¨1+(m2+m3)q¨2+m3q¨3+2(m2+m3)ΩU¯q˙1+c2q˙2+(m2+m3)Ω2U¯q1+k2q2=F(t)(m2+m3)Ω2U¯,E62
m3U¯q¨1+m3q¨2+m3q¨3+2m3ΩU¯q˙1+c3q˙3+m3Ω2U¯q1+k3q3=F(t)m3Ω2U¯.E63

In most cases, the force F(t)can be approximately a periodic function of the time or a constant. Thus, Eqs. (61)-(63) form a set of linear differential equations with periodic coefficients. Finally, the linearized differential equations of vibration can be expressed in the compact matrix form as

M(Ωt)q¨+C(Ωt)q˙+K(Ωt)q=d(Ωt),E64

where

M(Ωt)=[I1+(m2+m3)U¯2(m2+m3)U¯m3U¯(m2+m3)U¯(m2+m3)m3m3U¯m3m3]C(Ωt)=[c1+2(m2+m3)ΩU¯U¯002(m2+m3)ΩU¯c202m3ΩU¯0c3]K(Ωt)=[k1+FU¯+(m2+m3)Ω2(U¯U¯+U¯2)00(m2+m3)Ω2U¯k20m3Ω2U¯0k3]d(Ωt)=[FU¯(m2+m3)Ω2U¯U¯F(m2+m3)Ω2U¯Fm3Ω2U¯],   q=[q1q2q3].

We consider now the functionU(φ), called the first grade of the transmission function U(φ),where the angle φis the rotating angle of the cam shaft. In steady state motion of the cam mechanism, function U(φ)can be approximately expressed by a truncated Fourier series

U(φ)=k=1K(akcoskφ+bksinkφ).E65
 Parameters Units Values m2 (kg) 28 m3 (kg) 50 I1 (kgm2) 0.12 k1 (Nm/rad) 8×104 k2 ( N/m) 8.2×108 k3 ( N/m) 2.6×108 c1 (Nms/rad) 18.5 c2 (Ns/m) 1400 c3 (Ns/m) 1200

### Table 1.

Calculation parameters.

The functions U¯,U¯,U¯in Eq. (64) can then be calculated using Eq. (65) for φ=Ωt.Parameters used for the numerical calculation are listed in Table 1. Two set of coefficients akin Eq. (46) are given in Table 2 corresponding to two different cases of cam profile, coefficients bk=0.Without loss of generality, the external force Fis assumed to have a constant value of 100 N.

 ak( m) Case 1 Case 2 a1 0.22165 0.22206 a2 0 0 a3 0.05560 0.08539 a4 0 0 a5 - 0.01706 0.00518 a6 0 0 a7 0 - 0.00373 a8 0 0 a9 0 0.00345 a10 0 0 a11 0 - 0.00182 a12 0 0

### Table 2.

Fourier coefficients akofU(φ).

The rotating speed of the driver input nintakes firstly the value of 100 (rpm) corresponding to angular velocity Ω10.47(rad/s) for the calculation. The periodic solutions of Eq. (64) are then calculated using the numerical procedures proposed in Section 3. The results of a periodic solution for coordinate q3, which represents the dynamic transmission errors within the considered system, are shown in Figures 3 and 4. The influence of cam profile to the vibration response of the system can be recognized by a considerable difference in the vibration amplitude of both curves in Figure 3 and the frequency content of spectrums in Figure 4. In addition, the spectrums in Figure 4 shows harmonic components of the rotating frequency, such as ,Ω 3Ω, 5Ω which indicate stationary periodic vibrations.

Figures 5 and 6 show the calculating results with rotating speed nin=600(rpm), corresponding to Ω62.8(rad/s). The mechanism has a more serious dynamic transmission error at high speeds. It can be seen clearly from the frequency spectrums that the steady state vibration at high speeds of the considered cam mechanism may include tens harmonics of the rotating frequency as mentioned in [3].

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.

### 4.2. Parametric vibration of a gear - pair system with faulted meshing

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].

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 kz(t)is expressed as a time-varying function. The gear-pair is assumed to operate under high torque condition with zero backlash and the effect of friction forces at the meshing interface is neglected. The viscous damping coefficient of the gear mesh czis assumed to be constant. The differential equations of motion for this system can be expressed in the form

J1φ¨1+rb1kz(t)[rb1φ1+rb2φ2+e(t)]+rb1cz[rb1φ˙1+rb2φ˙2+e˙(t)]=M1(t),E66
J2φ¨2+rb2kz(t)[rb1φ1+rb2φ2+e(t)]+rb2cz[rb1φ˙1+rb2φ˙2+e˙(t)]=M2(t).E67

where φi,φ˙i,φ¨i(i= 1,2) are rotation angle, angular velocity, angular acceleration of the input pinion and the output wheel respectively. J1 and J2 are the mass moments of inertia of the gears. M1(t) and M2(t) denote the external torques load applied on the system. rb1 and rb2 represent the base radii of the gears. By introducing the composite coordinate

q=rb1φ1+rb2φ2.E68

Eqs. (66) and (67) yield a single differential equation in the following form

mredq¨+kz(t)q+czq˙=F(t)kz(t)e(t)cze˙(t),E69

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 φ˙1=ω1=const,φ˙2=ω2=const,cz=0,kz(t)=k0,the transmission error qis equal to the static tooth deflection under constant load q0as q=rb1φ1+rb2φ2=q0.Eq. (69) yields the following relation

F(t)F0(t)=k0q0+k0e(t).E71

Eq. (69) can then be rewritten in the form

mredq¨+kz(t)q+czq˙f(t)=0,E72

wheref(t)=k0q0[kz(t)k0]e(t)cze˙(t).

In steady state motion of the gear system, the mesh stiffness kz(t) can be approximately represented by a truncated Fourier series [33]

kz(t)=k0+n=1Nkncos(nωzt+γn).E73

where ωzis the gear meshing angular frequency which is equal to the number of gear teeth times the shaft angular frequency and Nis the number of terms of the series.

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

e(t)=i=1Ieicos(iω1t+αi).E74
 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

### Table 3.

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 includeJ1= 0.093(kgm2), J2 = 0.272 (kgm2) and nominal pinion speed of 1800 rpm (f1 = 30 Hz). The mesh stiffness of the test gear pair at particular meshing position was obtained by means of a FEM software [29]. The static tooth deflection is estimated to be q0 = 1.2×10-5(m). The values of Fourier coefficients of the mesh stiffness with corresponding phase angles are given in Table 4. The mean value of the undamped natural frequency ω¯0=k0/mred5462s-1,corresponding to f¯0=ω¯0/2π869(Hz). Based on the experimental work, the mean value of the Lehr damping ratio ζ¯=0.024is used for the dynamic model. The damping coefficient czcan then be determined by cz=2ω¯0ζ¯mred.

 n kn(N/m) γn(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

### Table 4.

Fourier coefficients and phase angles of the mesh stiffness.

 i Case 1 Case 2 ei(mm) αi(rad) ei(mm) αi(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

### Table 5.

Fourier coefficients and phase angles of excitation functione(t).

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 e(t)given in Table 5. The spectra in Figures 10(a) and 10(b) show clearly the meshing frequency and its harmonics with sideband structures. As expected, the sidebands are spaced by the rotational frequency f1 of the pinion. By comparing amplitude of these sidebands in both spectra, it can be concluded that the excitation function e(t) caused by tooth errors is responsible for generating sidebands.

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 fz 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 φ˙1and φ˙2)in order to determine experimentally the dynamic transmission error. The measurement was taken with two non-contacting transducers mounted in proximity to the shafts, positioned at the closest position to the test gears. The vibration signals were sampled at 10 kHz. The signal used in this study was recorded at the end of 12-hours total test time, at that time a surface fatigue failure occurred on some teeth of the pinion.

Figure 12 shows a frequency spectrum of the first derivative of the dynamic transmission error q˙(t)determined from the experimental data. The spectrum presents sidebands at the meshing frequency and its harmonics. In particular, the dominant sidebands are spaced by the rotational frequency of the pinion and characterized by high amplitude. This gives a clear indication of the presence of the faults on the pinion. By comparing the spectra displayed in Figures 13 and 14, it can be observed that the vibration spectrum calculated by numerical methods (Figure 13) and the spectrum of the measured vibration signal (Figure 14) show the same sideband structures.

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.

### 4.3. Periodic vibration of the transport manipulator of a forging press

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.

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 kiand constant damping coefficients ci,i=1,2.The rotating components are modeled by two rotating disks with moments of inertia I0and I1.The cam mechanism has a nonlinear transmission function U(φ1)as a function of the rotating angle φ1of the cam shaft, the driving torque from the motor M(t) and the external load F(t) applied on the system.

When the angular velocity Ωof the driver input is assumed to be constant in the steady state

φ0=Ωt,E75

one leads to the following relation

φ1=Ωt+q1E76

where q1is the difference between rotating angles φ0and φ1due to the presence of elastic element k1and damping element c1,resulted from the flexible transmission mechanism.

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

M(Ωt)q¨+C(Ωt)q˙+K(Ωt)q=d(Ωt)E77

where

M(Ωt)=[I1+m2U¯2m2U¯m2U¯m2],C(Ωt)=[c1+2m2ΩU¯U¯02m2ΩU¯c2]K(Ωt)=[k1+FU¯+m2Ω2(U¯U¯+U¯2)0m2Ω2U¯k2],d=[FU¯m2Ω2U¯U¯Fm2Ω2U¯],q=[q1q2]

In steady state motion of the cam mechanism, function U(φ)takes the form [18, 23]

U(φ)=k=1K(akcoskφ+bksinkφ)E78

The functions U¯,U¯,U¯in Eq. (77) can then be calculated using Eq. (78) for φ=Ωt.

The following parameters are used for numerical calculations: Rotating speed of the driver input n=50(rpm)corresponding to Ω=5.236(1/s),stiffness k1=7692Nm; k2=106N/m, damping coefficients c1=18.5Nms; c2=2332Ns/m, I1=1.11kgm2 and m2=136kg.

The Fourier coefficients akin Eq. (78) with K= 12 are given in Table 2 for two different cases and coefficients bk=0.We consider only periodic vibrations which are a commonly observed phenomenon in the system. The periodic solutions of Eq. (77) can be obtained by choosing appropriate initial conditions for the vector of variables q.

To verify the dynamic stable condition of the vibration system, the maximum of absolute value |ρ|maxof the solutions of the characteristic equation, according to Eq. (10), is now calculated. The obtained values for both cases are |ρ|max=0.001992(case 1) and |ρ|max=0.001623(case 2). It can be concluded that the system is dynamically stable for both two cases since |ρ|max<1.

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.

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.

## 5. Concluding remarks

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.

## Acknowledgments

This study was completed with the financial support by the Vietnam National Foundation for Science and Technology Development (NAFOSTED).

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Nguyen Van Khang and Nguyen Phong Dien (October 2nd 2012). Parametric Vibration Analysis of Transmission Mechanisms Using Numerical Methods, Advances in Vibration Engineering and Structural Dynamics, Francisco Beltran-Carbajal, IntechOpen, DOI: 10.5772/51157. Available from:

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