**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 mecha‐ nisms 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 theoreti‐ cally 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 *ki* and constant damping coefficients *ci* for *i* =1, 2, 3. Among them *k*<sup>1</sup> is the torsional stiffness of the cam shaft. Parame‐ ter *k*2 is 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 *k*3 denotes 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 *I*0 and *I*1. Let us in‐ troduce into our dynamic model the nonlinear transmission function *U* (*φ*1) of the cam mechanism as a function of the rotating angle *φ*1 of the cam shaft, the driving torque from the motor *M*(*t*) and the external load *F*(*t*) applied on the system.

**Figure 2.** Dynamic model of the cam mechanism.

Based on the proposed numerical procedures in this section, a computer program with MATLAB to calculate periodic vibrations of transmission mechanisms has been developed

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 mecha‐ nisms 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 theoreti‐

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 *ki* and constant damping

ter *k*2 is 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 *k*3 denotes

for *i* =1, 2, 3. Among them *k*<sup>1</sup> is the torsional stiffness of the cam shaft. Parame‐

and tested by the following application examples.

310 Advances in Vibration Engineering and Structural Dynamics

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

**4. Application examples**

cally and experimentally.

**Figure 1.** A cam mechanism.

coefficients *ci*

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

$$T = \frac{1}{2} I\_0 \dot{\rho}\_0^2 + \frac{1}{2} I\_1 \dot{\phi}\_1^2 + \frac{1}{2} m\_2 \dot{y}\_2^2 + \frac{1}{2} m\_3 \dot{y}\_3^2 \tag{41}$$

$$
\Pi = \frac{1}{2}k\_1(\varphi\_1 - \varphi\_0)^2 + \frac{1}{2}k\_2(y\_2 - y\_1)^2 + \frac{1}{2}k\_3(y\_3 - y\_2)^2 \tag{42}
$$

$$\mathbf{OP} = \frac{1}{2}c\_1(\dot{\varphi}\_1 - \dot{\varphi}\_0)^2 + \frac{1}{2}c\_2(\dot{y}\_2 - \dot{y}\_1)^2 + \frac{1}{2}c\_3(\dot{y}\_3 - \dot{y}\_2)^2\tag{43}$$

The virtual work done by all non-conservative forces is

$$
\sum \delta A = M \left( t \right) \delta q\_0 - F \left( t \right) \delta y\_3 \tag{44}
$$

Using the generalized coordinates *φ*0, *φ*1, *q*2, *q*3, we obtain the following relations

$$y\_1 = \mathcal{U}\left(\varphi\_1\right),\ y\_2 = y\_1 + q\_{2^s},\ y\_3 = y\_2 + q\_3\tag{45}$$

*φ*<sup>1</sup> =*Ωt* + *q*1, (56)

Parametric Vibration Analysis of Transmission Mechanisms Using Numerical Methods

<sup>2</sup> + …, (57)

http://dx.doi.org/10.5772/51157

313

<sup>2</sup> + …, (58)

<sup>2</sup> + …. (59)

(*Ωt*), *<sup>U</sup>*¯‴ <sup>=</sup>*<sup>U</sup>* ‴(*Ωt*). (60)

*<sup>U</sup>*¯″ *<sup>q</sup>*˙ <sup>1</sup>

*U*¯′ *U*¯″

, (61)

. (63)

, (62)

*U*¯″

where *q*1 is the difference between rotating angles *φ*0 and *φ*<sup>1</sup> due to the presence of the spring element *k*1 and the damping element *c*1. Assuming that *φ*1 varies little from its mean value during the steady-state motion, the transmission function *y*<sup>1</sup> =*U* (*φ*1) depends essen‐

> *q*<sup>1</sup> + 1 2 *U*¯″ *q*1

> > *q*<sup>1</sup> + 1 <sup>2</sup> *<sup>U</sup>*¯‴*q*<sup>1</sup>

> > > 1 2 *U*¯(4) *q*1

<sup>3</sup> <sup>+</sup> *<sup>c</sup>*<sup>1</sup> <sup>+</sup> 2(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*ΩU*¯′

<sup>−</sup>(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>Ω</sup>* <sup>2</sup>

*q*˙ 1

*<sup>U</sup>*¯‴*q*<sup>1</sup> <sup>+</sup> *<sup>k</sup>*3*q*<sup>3</sup> <sup>=</sup> <sup>−</sup> *<sup>F</sup>* (*t*)−*m*3*<sup>Ω</sup>* <sup>2</sup>

*U*¯″

) *<sup>q</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup> *<sup>F</sup>* (*t*)*<sup>U</sup>*¯′

*<sup>U</sup>*¯‴*q*<sup>1</sup> <sup>+</sup> *<sup>k</sup>*2*q*<sup>2</sup> <sup>=</sup> <sup>−</sup> *<sup>F</sup>*(*t*)−(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>Ω</sup>* <sup>2</sup>

<sup>3</sup> <sup>+</sup> 2(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*ΩU*¯″

*M* (*Ωt*)*q***¨** + *C*(*Ωt*)*q***˙** + *K*(*Ωt*)*q* =*d*(*Ωt*), (64)

tially on the input angle *φ*<sup>0</sup> =*Ωt*. Using the Taylor series expansion around *Ωt*, we get

(*Ω<sup>t</sup>* <sup>+</sup> *<sup>q</sup>*1)=*<sup>U</sup>*¯′ <sup>+</sup> *<sup>U</sup>*¯″

(*Ω<sup>t</sup>* <sup>+</sup> *<sup>q</sup>*1)=*<sup>U</sup>*¯″ <sup>+</sup> *<sup>U</sup>*¯‴*q*<sup>1</sup> <sup>+</sup>

(*Ωt*), *<sup>U</sup>*¯″ <sup>=</sup>*<sup>U</sup>* ″

Since the system performs small vibrations, i.e. there are only small vibrating amplitudes *q*1, *q*2 and *q*3, substituting Eqs. (57)-(59) into Eqs. (52)-(54) and neglecting nonlinear terms,

*<sup>U</sup>* (*φ*1)=*<sup>U</sup>* (*Ω<sup>t</sup>* <sup>+</sup> *<sup>q</sup>*1)=*<sup>U</sup>*¯ <sup>+</sup> *<sup>U</sup>*¯′

=*U* ′

we obtain the linear differential equations of vibration for the system

<sup>1</sup> + (*m*<sup>2</sup> + *m*3)*q*

*q* ¨ <sup>2</sup> <sup>+</sup> *<sup>m</sup>*3*<sup>U</sup>*¯′ *q* ¨

*<sup>U</sup>*¯‴ <sup>+</sup> *<sup>U</sup>*¯″<sup>2</sup>

¨ <sup>2</sup> + *m*3*q* ¨

*<sup>q</sup>*˙ <sup>1</sup> <sup>+</sup> *<sup>c</sup>*3*q*˙ <sup>3</sup> <sup>+</sup> *<sup>m</sup>*3*<sup>Ω</sup>* <sup>2</sup>

In most cases, the force *F* (*t*) can be approximately a periodic function of the time or a con‐ stant. Thus, Eqs. (61)-(63) form a set of linear differential equations with periodic coeffi‐ cients. Finally, the linearized differential equations of vibration can be expressed in the

<sup>1</sup> <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>*¯′

*U* ′

*U* ″

where we used the notations

(*I*<sup>1</sup> <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>*¯′<sup>2</sup>

*m*3*U*¯′ *q* ¨ <sup>1</sup> + *m*3*q* ¨ <sup>2</sup> + *m*3*q* ¨

where

compact matrix form as

(*φ*1)=*<sup>U</sup>* ′

(*φ*1)=*<sup>U</sup>* ″

*<sup>U</sup>*¯ <sup>=</sup>*<sup>U</sup>* (*Ωt*), *<sup>U</sup>*¯′

)*q* ¨

> *q* ¨

<sup>3</sup> <sup>+</sup> <sup>2</sup>*m*3*ΩU*¯″

<sup>+</sup>*c*2*q*˙ <sup>2</sup> <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>Ω</sup>* <sup>2</sup>

<sup>+</sup> *<sup>k</sup>*<sup>1</sup> <sup>+</sup> *<sup>F</sup>* (*t*)*<sup>U</sup>*¯″ <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>Ω</sup>* 2(*U*¯′

(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>*¯′

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

$$T = \frac{1}{2} I\_0 \dot{\rho}\_0^2 + \frac{1}{2} I\_1 \dot{\rho}\_1^2 + \frac{1}{2} m\_2 \{\mathcal{U}^\prime \dot{\rho}\_1 + \dot{\eta}\_2\}^2 + \frac{1}{2} m\_3 \{\mathcal{U}^\prime \dot{\rho}\_1 + \dot{\eta}\_2 + \dot{\eta}\_3\}^2,\tag{46}$$

$$
\Pi = \frac{1}{2}k\_1(q\_1 - q\_0)^2 + \frac{1}{2}k\_2q\_2^2 + \frac{1}{2}k\_3q\_3^2 \tag{47}
$$

$$\mathcal{O} = \frac{1}{2}c\_1(\dot{\varphi}\_1 - \dot{\varphi}\_0)^2 + \frac{1}{2}c\_2\dot{\eta}\_2^{\prime} + \frac{1}{2}c\_3\dot{\eta}\_3^{\prime},\tag{48}$$

$$
\sum \delta A = M \left( t \right) \delta q\_0 - F \left( t \right) L^\prime \delta q\_1 - F \left( t \right) \delta q\_2 - F \left( t \right) \delta q\_3 \tag{49}
$$

where the prime represents the derivative with respect to the generalized coordinate *φ*1. The generalized forces of all non-conservative forces are then derived from Eq. (49) as

$$\left.Q\_{\varphi\_0}^{\*} = M\left(t\right)\_{\prime}\right.\left.Q\_{\varphi\_1}^{\*} = -F\left(t\right)\mathcal{U}\left.\right.\right.\left.Q\_{\varphi\_2}^{\*} = -F\left(t\right)\_{\prime}\right.\left.Q\_{\varphi\_3}^{\*} = -F\left(t\right).\tag{50}$$

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 *φ*0, *φ*1, *q*2, *q*<sup>3</sup>

$$I\_0 \ddot{\phi}\_0 - c\_1(\phi\_1 - \phi\_0) - k\_1(\phi\_1 - \phi\_0) = M \text{ (t)},\tag{51}$$

$$\begin{aligned} \left[I\_1 + (m\_2 + m\_3)\mathbf{U}\right]^\prime \stackrel{\textstyle T}{\mathbf{I}} \stackrel{\textstyle T}{\mathbf{I}} & \mathbf{0} + (m\_2 + m\_3)\mathbf{U}\stackrel{\textstyle T}{\mathbf{I}} \stackrel{\textstyle T}{\mathbf{I}}\_2 + m\_3\mathbf{U}\stackrel{\textstyle T}{\mathbf{I}} \stackrel{\textstyle T}{\mathbf{I}}\_3 + (m\_2 + m\_3)\mathbf{U}\stackrel{\textstyle T}{\mathbf{I}}\stackrel{\textstyle T}{\mathbf{I}}\mathbf{I}\stackrel{\textstyle T}{\mathbf{I}}\mathbf{I}\tag{52} \\ + c\_1(\dot{\boldsymbol{\phi}}\_1 - \dot{\boldsymbol{\phi}}\_0) + k\_1(\boldsymbol{\phi}\_1 - \boldsymbol{\phi}\_0) &= -F(t)\mathbf{U}\stackrel{\textstyle \prime}{\mathbf{I}}\tag{52} \end{aligned} \tag{52}$$

$$\left( (m\_2 + m\_3) \text{Li}^{\prime} \ddot{\hat{\boldsymbol{\varphi}}}\_1 + (m\_2 + m\_3) \ddot{\hat{\boldsymbol{q}}}\_2 + m\_3 \ddot{\hat{\boldsymbol{q}}}\_3 + (m\_2 + m\_3) \text{Li}^{\prime} \dot{\hat{\boldsymbol{\varphi}}}\_1 \dot{\hat{\boldsymbol{\varphi}}}\_2 + c\_2 \dot{\hat{\boldsymbol{q}}}\_2 + k\_2 \dot{\boldsymbol{q}}\_2 = -F \text{(t)},\tag{53}$$

$$m\_3 \text{Li}^"\ddot{\boldsymbol{\phi}}\_1 + m\_3 \ddot{\boldsymbol{q}}\_2 + m\_3 \ddot{\boldsymbol{q}}\_3 + m\_3 \text{Li}^"\dot{\boldsymbol{\phi}}\_1 \dot{\boldsymbol{\phi}}\_2 + c\_3 \dot{\boldsymbol{q}}\_3 + k\_3 \boldsymbol{q}\_3 = -F(t). \tag{54}$$

When the angular velocity *Ω* of the driver input is assumed to be constant in the steady state

$$
\varphi\_0 = \Omega t\_\prime \tag{55}
$$

one leads to the following relation

Parametric Vibration Analysis of Transmission Mechanisms Using Numerical Methods http://dx.doi.org/10.5772/51157 313

$$
\varphi\_1 = \Omega t + q\_{1\nu} \tag{56}
$$

where *q*1 is the difference between rotating angles *φ*0 and *φ*<sup>1</sup> due to the presence of the spring element *k*1 and the damping element *c*1. Assuming that *φ*1 varies little from its mean value during the steady-state motion, the transmission function *y*<sup>1</sup> =*U* (*φ*1) depends essen‐ tially on the input angle *φ*<sup>0</sup> =*Ωt*. Using the Taylor series expansion around *Ωt*, we get

$$\mathcal{U}\mathcal{U}(\boldsymbol{\varphi}\_{1}) = \mathcal{U}\left(\mathcal{Q}\boldsymbol{t} + \boldsymbol{q}\_{1}\right) = \overline{\mathcal{U}} + \overline{\mathcal{U}}\,^{\prime}\boldsymbol{q}\_{1} + \frac{1}{2}\overline{\mathcal{U}}\,^{\prime}\boldsymbol{q}\_{1}\,^{2} + \dots \,^{\prime} \tag{57}$$

$$\text{III}^{'}(\varphi\_1) = \text{II}^{'}(\Omega t + q\_1) = \overline{\text{II}}^{'} + \overline{\text{II}}^{''} q\_1 + \frac{1}{2} \overline{\text{II}}^{'''} q\_1^2 + \dots \tag{58}$$

$$\text{III}^{''}(\phi\_1) = \text{II}^{''}(\Omega t + q\_1) = \overline{\text{II}}^{''} + \overline{\text{II}}^{''} q\_1 + \frac{1}{7} \overline{\text{II}}^{(4)} q\_1 2 + \dots \tag{59}$$

where we used the notations

Using the generalized coordinates *φ*0, *φ*1, *q*2, *q*3, we obtain the following relations

*<sup>φ</sup>*˙ <sup>1</sup> <sup>+</sup> *<sup>q</sup>*˙ 2)<sup>2</sup> <sup>+</sup>

1 <sup>2</sup> *k*2*q*<sup>2</sup> 2 + 1 <sup>2</sup> *k*3*q*<sup>3</sup>

1 <sup>2</sup> *c*2*q*˙ <sup>2</sup> 2 + 1 <sup>2</sup> *c*3*q*˙ <sup>3</sup>

where the prime represents the derivative with respect to the generalized coordinate *φ*1. The

, *Qq*<sup>2</sup>

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

> *q* ¨

<sup>3</sup> <sup>+</sup> *<sup>m</sup>*3*<sup>U</sup>* ″

<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3*<sup>U</sup>* ′ *q* ¨

<sup>3</sup> <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>* ″

*φ*˙ 1

When the angular velocity *Ω* of the driver input is assumed to be constant in the steady state

\* <sup>=</sup> <sup>−</sup> *<sup>F</sup>* (*t*), *Qq*<sup>3</sup>

<sup>0</sup> −*c*1(*φ*˙ <sup>1</sup> −*φ*˙ <sup>0</sup>)−*k*1(*φ*<sup>1</sup> −*φ*0)=*M* (*t*), (51)

<sup>3</sup> <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>* ′

*φ*˙ 1

*U* ″ *φ*˙ 1 2

<sup>2</sup> + *c*3*q*˙ <sup>3</sup> + *k*3*q*<sup>3</sup> = − *F*(*t*). (54)

*φ*<sup>0</sup> =*Ωt*, (55)

<sup>2</sup> + *c*2*q*˙ <sup>2</sup> + *k*2*q*<sup>2</sup> = − *F*(*t*), (53)

, (52)

generalized forces of all non-conservative forces are then derived from Eq. (49) as

\* = − *F* (*t*)*U* ′

<sup>1</sup> <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>* ′

1 <sup>2</sup> *<sup>m</sup>*3(*<sup>U</sup>* ′

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

312 Advances in Vibration Engineering and Structural Dynamics

*<sup>Π</sup>* <sup>=</sup> <sup>1</sup>

*<sup>Φ</sup>* <sup>=</sup> <sup>1</sup>

\* <sup>=</sup>*<sup>M</sup>* (*t*), *<sup>Q</sup>φ*<sup>1</sup>

*I*0*φ*¨

<sup>+</sup>*c*1(*φ*˙ <sup>1</sup> <sup>−</sup>*φ*˙ <sup>0</sup>) <sup>+</sup> *<sup>k</sup>*1(*φ*<sup>1</sup> <sup>−</sup>*φ*0)= <sup>−</sup> *<sup>F</sup>* (*t*)*<sup>U</sup>* ′

<sup>1</sup> + (*m*<sup>2</sup> + *m*3)*q*

¨ <sup>2</sup> + *m*3*q* ¨

*φ*¨

∑ *<sup>δ</sup>A*=*<sup>M</sup>* (*t*)*δφ*<sup>0</sup> <sup>−</sup> *<sup>F</sup>* (*t*)*<sup>U</sup>* ′

<sup>2</sup> *k*1(*φ*<sup>1</sup> −*φ*0)<sup>2</sup> +

<sup>2</sup> *c*1(*φ*˙ <sup>1</sup> −*φ*˙ 0)<sup>2</sup> +

*<sup>T</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> *I*0*φ*˙ <sup>0</sup> 2 + 1 <sup>2</sup> *I*1*φ*˙ <sup>1</sup> 2 + 1 <sup>2</sup> *<sup>m</sup>*2(*<sup>U</sup>* ′

*Qφ*<sup>0</sup>

*<sup>I</sup>*<sup>1</sup> <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>* ′<sup>2</sup>

*φ*¨

*<sup>m</sup>*3*<sup>U</sup>* ′ *φ*¨ <sup>1</sup> + *m*3*q* ¨ <sup>2</sup> + *m*3*q* ¨

one leads to the following relation

(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>* ′

*φ*0, *φ*1, *q*2, *q*<sup>3</sup>

*y*<sup>1</sup> =*U* (*φ*1), *y*<sup>2</sup> = *y*<sup>1</sup> + *q*2, *y*<sup>3</sup> = *y*<sup>2</sup> + *q*<sup>3</sup> (45)

*<sup>φ</sup>*˙ <sup>1</sup> <sup>+</sup> *<sup>q</sup>*˙ <sup>2</sup> <sup>+</sup> *<sup>q</sup>*˙ <sup>3</sup>)<sup>2</sup>

*δφ*<sup>1</sup> − *F* (*t*)*δq*<sup>2</sup> − *F* (*t*)*δq*3, (49)

, (46)

2, (47)

2, (48)

\* = − *F*(*t*). (50)

$$
\overline{\boldsymbol{\Pi}} = \boldsymbol{\Pi} \left( \boldsymbol{\Omega} t \right), \ \overline{\boldsymbol{\Pi}}' = \boldsymbol{\Pi} \left( \boldsymbol{\Omega} t \right), \ \overline{\boldsymbol{\Pi}}'' = \boldsymbol{\Pi} \left( \boldsymbol{\Omega} t \right), \ \overline{\boldsymbol{\Pi}}'' = \boldsymbol{\Pi} \left( \boldsymbol{\Omega} t \right). \tag{60}
$$

Since the system performs small vibrations, i.e. there are only small vibrating amplitudes *q*1, *q*2 and *q*3, substituting Eqs. (57)-(59) into Eqs. (52)-(54) and neglecting nonlinear terms, we obtain the linear differential equations of vibration for the system

$$\begin{aligned} \left[ \left( \mathbf{I}\_1 + (m\_2 + m\_3)\overline{\mathbf{U}}\prime\prime \right) \overline{\mathbf{q}}\_1 + (m\_2 + m\_3)\overline{\mathbf{U}}\prime \overline{\mathbf{q}}\_2 + m\_3\overline{\mathbf{U}}\prime \overline{\mathbf{q}}\_3 + \left[ \mathbf{c}\_1 + 2(m\_2 + m\_3)\Omega\prime \overline{\mathbf{U}}\prime\prime \overline{\mathbf{U}}\prime \right] \overline{\mathbf{q}}\_1 \\ + \left[ \mathbf{k}\_1 + F(t)\overline{\mathbf{U}}\prime\prime + (m\_2 + m\_3)\Omega\prime^2 \overline{\left(\overline{\mathbf{U}}\prime\prime \overline{\mathbf{U}}\prime\prime + \overline{\mathbf{U}}\prime\prime^2\right)} \right] \overline{\mathbf{q}}\_1 = -F(t)\overline{\mathbf{U}}\prime - (m\_2 + m\_3)\Omega\prime^2 \overline{\mathbf{U}}\prime \overline{\mathbf{U}}\prime, \end{aligned} \tag{61}$$

$$\begin{aligned} (m\_2 + m\_3)\overline{\boldsymbol{\Omega}}^{\prime}\overline{\boldsymbol{q}}\_1 + (m\_2 + m\_3)\overline{\boldsymbol{q}}\_2 + m\_3\overline{\boldsymbol{q}}\_3 + 2(m\_2 + m\_3)\boldsymbol{\Omega}\,\overline{\boldsymbol{\Omega}}^{\prime}\dot{\boldsymbol{q}}\_1 \\ + c\_2\dot{\boldsymbol{q}}\_2 + (m\_2 + m\_3)\boldsymbol{\Omega}\,^2\overline{\boldsymbol{\Omega}}^{\prime}\boldsymbol{q}\_1 + k\_2\boldsymbol{q}\_2 &= -\boldsymbol{F}(t) - (m\_2 + m\_3)\boldsymbol{\Omega}\,^2\overline{\boldsymbol{\Omega}}^{\prime}\boldsymbol{f}\_2 \end{aligned} \tag{62}$$

$$2m\_3\overline{\boldsymbol{\Pi}}\prime\prime\prime\_1 + m\_3\overline{\boldsymbol{q}}\_2 + m\_3\overline{\boldsymbol{q}}\_3 + 2m\_3\boldsymbol{\Omega}\prime\overline{\boldsymbol{\Pi}}\prime\prime\_1 + c\_3\dot{\boldsymbol{q}}\_3 + m\_3\boldsymbol{\Omega}\prime^2\overline{\boldsymbol{\Pi}}\prime\prime\_1 + k\_3\boldsymbol{q}\_3 = -\boldsymbol{F}\prime(t) - m\_3\boldsymbol{\Omega}\prime^2\overline{\boldsymbol{\Pi}}\prime.\tag{63}$$

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

$$\stackrel{\circ}{M}(\varOmega t)\ddot{\mathfrak{q}} + \stackrel{\circ}{C}(\varOmega t)\dot{\mathfrak{q}} + \stackrel{\circ}{K}(\varOmega t)\mathfrak{q} = \stackrel{\circ}{d}(\varOmega t),\tag{64}$$

where

$$\begin{split} \mathbf{M} \begin{bmatrix} \mathbf{M} \end{bmatrix} &= \begin{bmatrix} \begin{matrix} I\_1 + (m\_2 + m\_3)\overline{\mathbf{U}} \ \overline{\mathbf{U}} \end{matrix} & (m\_2 + m\_3)\overline{\mathbf{U}} \ \overline{\mathbf{U}} & m\_3\overline{\mathbf{U}} \\ (m\_2 + m\_3)\overline{\mathbf{U}} \ \overline{\mathbf{U}} & (m\_2 + m\_3) & m\_3 \end{bmatrix} \mathbf{C} \begin{bmatrix} c\_1 + 2(m\_2 + m\_3)\Omega\overline{\mathbf{U}} \ \overline{\mathbf{U}} \ \overline{\mathbf{U}} & \mathbf{0} \\ 2(m\_2 + m\_3)\Omega\overline{\mathbf{U}} & c\_2 & 0 \\ m\_3\Omega\overline{\mathbf{U}} & 0 & c\_3 \end{bmatrix} \\ \begin{bmatrix} k\_1 + F\overline{\mathbf{U}} \ \overline{\mathbf{U}} \ \overline{\mathbf{U}} \ \overline{\mathbf{U}} & (m\_2 + m\_3)\Omega\overline{\mathbf{U}} \ \overline{\mathbf{U}} & \overline{\mathbf{U}} \ 0 & 0 \\ (m\_2 + m\_3)\Omega\overline{\mathbf{U}} & k\_2 & 0 \\ m\_3\Omega\overline{\mathbf{U}} & 0 & k\_3 \end{bmatrix} \\ \begin{bmatrix} -F\overline{\mathbf{U}} \ -(m\_2 + m\_3)\Omega\overline{\mathbf{U}} \ \overline{\mathbf{U}} \ \overline{\mathbf{U}} \\ -F - (m\_2 + m\_3)\Omega\overline{\mathbf{U}} \ \overline{\mathbf{U}} \ \overline{\mathbf{U}} \\ -F - (m\_2 + m\_3)\Omega\overline{\mathbf{U}} \ \overline{\mathbf{U}} \end{bmatrix} \end{split}$$

We consider now the function *U* ′ (*φ*) , called the first grade of the transmission function *U* (*φ*), where the angle *φ* is 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


$$\operatorname{LLI}^{'}(\varphi) = \sum\_{k=1}^{K} (a\_k \cos k\varphi + b\_k \sin k\varphi). \tag{65}$$

*ak* **( m) Case 1 Case 2** *a* <sup>1</sup> 0.22165 0.22206

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*a* <sup>3</sup> 0.05560 0.08539

*a* <sup>5</sup> - 0.01706 0.00518

*a* <sup>7</sup> 0 - 0.00373

*a* <sup>9</sup> 0 0.00345

*a* <sup>11</sup> 0 - 0.00182

*a* <sup>2</sup> 0 0

*a* <sup>4</sup> 0 0

*a* <sup>6</sup> 0 0

*a* <sup>8</sup> 0 0

*a* <sup>10</sup> 0 0

*a* <sup>12</sup> 0 0

**Figure 3.** Dynamic transmission errors *q*3 with nim=100(rpm) for Case 1 (left) and Case 2 (right).

such as ,Ω 3Ω, 5Ω which indicate stationary periodic vibrations.

The rotating speed of the driver input *nin* takes firstly the value of 100 (rpm) corresponding to angular velocity *Ω* ≈10.47 (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 peri‐ odic solution for coordinate *q* 3, which represents the dynamic transmission errors within the considered system, are shown in Figures 3 and 4. The influence of cam profile to the vibra‐ tion 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,

(φ) .

**Table 2.** Fourier coefficients *ak* of *<sup>U</sup>* ′

**Table 1.** Calculation parameters.

The functions *<sup>U</sup>*¯′ , *U*¯′′ , *<sup>U</sup>*¯‴ in Eq. (64) can then be calculated using Eq. (65) for *<sup>φ</sup>* <sup>=</sup>*Ωt*. Param‐ eters used for the numerical calculation are listed in Table 1. Two set of coefficients *ak* in 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 *F* is assumed to have a constant value of 100 N.


**Table 2.** Fourier coefficients *ak* of *<sup>U</sup>* ′ (φ) .

*M* (*Ωt*)=

*K*(*Ωt*)=

*d*(*Ωt*)=

*<sup>I</sup>*<sup>1</sup> <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>*¯′<sup>2</sup>

(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>Ω</sup>* <sup>2</sup>

<sup>−</sup> *<sup>F</sup>* <sup>−</sup>(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>Ω</sup>* <sup>2</sup>

We consider now the function *U* ′

<sup>−</sup> *<sup>F</sup>* <sup>−</sup>*m*3*<sup>Ω</sup>* <sup>2</sup>

mechanism, function *U* ′

**Table 1.** Calculation parameters.

, *U*¯′′

The functions *<sup>U</sup>*¯′

*<sup>m</sup>*3*<sup>Ω</sup>* <sup>2</sup>

<sup>−</sup> *FU*¯′

*<sup>k</sup>*<sup>1</sup> <sup>+</sup> *FU*¯″ <sup>+</sup> (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>Ω</sup>* <sup>2</sup>

314 Advances in Vibration Engineering and Structural Dynamics

<sup>−</sup>(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>Ω</sup>* <sup>2</sup>

*U*¯″

(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>*¯′ (*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3) *<sup>m</sup>*<sup>3</sup> *<sup>m</sup>*3*<sup>U</sup>*¯′ *<sup>m</sup>*<sup>3</sup> *<sup>m</sup>*<sup>3</sup>

(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*<sup>U</sup>*¯′ *<sup>m</sup>*3*<sup>U</sup>*¯′

*<sup>U</sup>*¯‴ *<sup>k</sup>*<sup>2</sup> <sup>0</sup>

, *q* =

*<sup>U</sup>*¯‴ <sup>+</sup> *<sup>U</sup>*¯″<sup>2</sup>

*q*1 *q*2 *q*3 .

(*U*¯′

*<sup>U</sup>*¯‴ <sup>0</sup> *<sup>k</sup>*<sup>3</sup>

*U*¯′ *U*¯″

*U* ′

(*φ*)=∑ *k*=1 *K*

**Parameters Units Values** *m*<sup>2</sup> (kg) 28 *m*<sup>3</sup> (kg) 50 *I*<sup>1</sup> (kgm2) 0.12 *k*<sup>1</sup> (Nm/rad) 8×10<sup>4</sup> *k*<sup>2</sup> ( N/m) 8.2×10<sup>8</sup> *k*<sup>3</sup> ( N/m) 2.6×10<sup>8</sup> *c*<sup>1</sup> (Nms/rad) 18.5 *c*<sup>2</sup> (Ns/m) 1400 *c*<sup>3</sup> (Ns/m) 1200

*U*¯″

*C*(*Ωt*)=

) 0 0

where the angle *φ* is the rotating angle of the cam shaft. In steady state motion of the cam

*<sup>c</sup>*<sup>1</sup> <sup>+</sup> 2(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*ΩU*¯′

(*φ*) , called the first grade of the transmission function *U* (*φ*),

(*ak* cos*kφ* + *bk* sin*kφ*). (65)

(*φ*) can be approximately expressed by a truncated Fourier series

, *<sup>U</sup>*¯‴ in Eq. (64) can then be calculated using Eq. (65) for *<sup>φ</sup>* <sup>=</sup>*Ωt*. Param‐

eters used for the numerical calculation are listed in Table 1. Two set of coefficients *ak* in 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 *F* is assumed to have a constant value of 100 N.

2(*m*<sup>2</sup> <sup>+</sup> *<sup>m</sup>*3)*ΩU*¯″ *<sup>c</sup>*<sup>2</sup> <sup>0</sup> <sup>2</sup>*m*3*ΩU*¯″ <sup>0</sup> *<sup>c</sup>*<sup>3</sup>

*<sup>U</sup>*¯″ <sup>0</sup> <sup>0</sup>

**Figure 3.** Dynamic transmission errors *q*3 with nim=100(rpm) for Case 1 (left) and Case 2 (right).

The rotating speed of the driver input *nin* takes firstly the value of 100 (rpm) corresponding to angular velocity *Ω* ≈10.47 (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 peri‐ odic solution for coordinate *q* 3, which represents the dynamic transmission errors within the considered system, are shown in Figures 3 and 4. The influence of cam profile to the vibra‐ tion 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.

**Figure 4.** Frequency spectrum of *q*3 with *nin* =100(*rpm*) for Case 1 (left) and Case 2 (right).

Figures 5 and 6 show the calculating results with rotating speed *nin* =600 (rpm), correspond‐ ing to *Ω* ≈62.8 (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 vibra‐ tion at high speeds of the considered cam mechanism may include tens harmonics of the ro‐ tating frequency as mentioned in [3].

**Figure 7.** Comparison of the computation time for the model of cam mechanism.

the excitation from gear transmission errors [31-33].

**Figure 8.** Dynamic model of the gear-pair system with faulted meshing.

responsible for generating such sidebands.

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

Dynamic modeling of gear vibrations offers a better understanding of the vibration genera‐ tion 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

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From experimental works, it is well known that the most important components in gear vi‐ bration 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 side‐ bands 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

**Figure 5.** Dynamic transmission errors *q*3 with *nin* =600(*rpm*) for Case 1.

**Figure 6.** Dynamic transmission errors *q*3 with *nin* =600(*rpm*) for Case 2.

The calculation of the periodic solution of Eq. (64) was implemented by a self-written com‐ puter 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 nu‐ merical procedures are identical, but the computation time with Newmark method is great‐ ly reduced in comparison with Runge-Kutta method as shown in Figure 7, especially in the cases of large number of time steps.

**Figure 7.** Comparison of the computation time for the model of cam mechanism.

**Figure 4.** Frequency spectrum of *q*3 with *nin* =100(*rpm*) for Case 1 (left) and Case 2 (right).

tating frequency as mentioned in [3].

316 Advances in Vibration Engineering and Structural Dynamics

**Figure 5.** Dynamic transmission errors *q*3 with *nin* =600(*rpm*) for Case 1.

**Figure 6.** Dynamic transmission errors *q*3 with *nin* =600(*rpm*) for Case 2.

cases of large number of time steps.

Figures 5 and 6 show the calculating results with rotating speed *nin* =600 (rpm), correspond‐ ing to *Ω* ≈62.8 (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 vibra‐ tion at high speeds of the considered cam mechanism may include tens harmonics of the ro‐

The calculation of the periodic solution of Eq. (64) was implemented by a self-written com‐ puter 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 nu‐ merical procedures are identical, but the computation time with Newmark method is great‐ ly reduced in comparison with Runge-Kutta method as shown in Figure 7, especially in the

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

Dynamic modeling of gear vibrations offers a better understanding of the vibration genera‐ tion 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].

**Figure 8.** Dynamic model of the gear-pair system with faulted meshing.

From experimental works, it is well known that the most important components in gear vi‐ bration 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 side‐ bands 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 *cz* is assumed to be constant. The differential equations of motion for this system can be expressed in the form

$$\left[J\_1\ddot{\boldsymbol{\phi}}\_1 + r\_{b1}k\_z(t)\right]\mathbf{r}\_{b1}\boldsymbol{\phi}\_1 + r\_{b2}\boldsymbol{\phi}\_2 + \dot{\boldsymbol{e}}(t)\mathbf{J} + r\_{b1}\mathbf{c}\_z\left[r\_{b1}\dot{\boldsymbol{\phi}}\_1 + r\_{b2}\dot{\boldsymbol{\phi}}\_2 + \dot{\boldsymbol{e}}(t)\right] = M\_1(t),\tag{66}$$

$$\left[\mathbf{J}\_{1}\ddot{\boldsymbol{\varphi}}\_{2} + r\_{b2}\dot{\boldsymbol{k}}\_{z}(t)\right]\mathbf{r}\_{b1}\boldsymbol{\varphi}\_{1} + r\_{b2}\boldsymbol{\varphi}\_{2} + \dot{\mathbf{e}}(t)\mathbf{J} + r\_{b2}\boldsymbol{\varepsilon}\_{2}\mathbf{I}\_{b1}\dot{\boldsymbol{\varphi}}\_{1} + r\_{b2}\dot{\boldsymbol{\varphi}}\_{2} + \dot{\mathbf{e}}(t)\mathbf{J}\right] = \mathbf{M}\_{2}(t).\tag{67}$$

where *φ<sup>i</sup>* , *φ*˙ *<sup>i</sup>* , *φ*¨ *<sup>i</sup>* (*i* = 1,2) are rotation angle, angular velocity, angular acceleration of the in‐ put pinion and the output wheel respectively. *J* 1 and *J* 2 are the mass moments of inertia of the gears. *M* 1(*t*) and *M* 2(*t*) denote the external torques load applied on the system. *r <sup>b</sup>*1 and *r <sup>b</sup>*2 represent the base radii of the gears. By introducing the composite coordinate

$$q = r\_{b1}q\_1 + r\_{b2}q\_2. \tag{68}$$

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

where *f* (*t*)=*k*0*q*<sup>0</sup> − *kz*(*t*)−*k*<sup>0</sup> *e*(*t*)−*cze*˙(*t*).

**Table 3.** Parameters of the test gears.

clude *J*1= 0.093 (kgm2

equation with the periodic coefficients.

resented by a truncated Fourier series [33]

*mred q*

*kz*(*t*)=*k*<sup>0</sup> <sup>+</sup> ∑

*n*=1 *N*

times the shaft angular frequency and *N* is the number of terms of the series.

errors are situated at the teeth of the pinion, *e*(*t*) may be taken in the form

*e*(*t*)=∑ *i*=1 *I ei*

Material steel Module (mm) 4.50 Pressure angle (o) 20.00 Helical angle (o) 14.56

), *J* <sup>2</sup> = 0.272 (kgm2

In steady state motion of the gear system, the mesh stiffness *k <sup>z</sup>*(*t*) can be approximately rep‐

where *ωz* is the gear meshing angular frequency which is equal to the number of gear teeth

In general, the error components are no identical for each gear tooth and will produce dis‐ placement 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

cos(*iω*1*t* + *α<sup>i</sup>*

**Parameters Pinion Wheel** Gear type helical, standard involute

Number of teeth z 14 39 face width (mm) 67.00 45.00 base circle radius (mm) 30.46 84.86

Therefore, the vibration equation of gear-pair system according to Eq. (72) is a differential

According to the experimental setup which will be described later, the model parameters in‐

The mesh stiffness of the test gear pair at particular meshing position was obtained by

¨ <sup>+</sup> *kz*(*t*)*<sup>q</sup>* <sup>+</sup> *czq*˙ <sup>−</sup> *<sup>f</sup>* (*t*)=0, (72)

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Parametric Vibration Analysis of Transmission Mechanisms Using Numerical Methods

*kn*cos(*nωzt* + *γn*). (73)

). (74)

) and nominal pinion speed of 1800 rpm (*f* <sup>1</sup> = 30 Hz).

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

$$
\dot{m}\_{rel}\ddot{\ddot{q}} + k\_z(t)q + c\_z\dot{q} = F(t) - k\_z(t)e(t) - c\_z\dot{e}(t),\tag{69}
$$

where

$$m\_{red} = \frac{J\_1 J\_2}{J\_1 r\_{b2}^2 + J\_2 r\_{b1}^2} \qquad F(t) = m\_{red} \Big( \frac{M\_1(t) r\_{b1}}{J\_1} + \frac{M\_2(t) r\_{b2}}{J\_2} \Big). \tag{70}$$

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*˙ <sup>1</sup> =*ω*<sup>1</sup> =*const*, *φ*˙ <sup>2</sup> =*ω*<sup>2</sup> =*const*, *cz* =0, *kz*(*t*)=*k*0, the transmission error *q* is equal to the static tooth deflection under constant load *q*0 as *q* =*rb*1*φ*<sup>1</sup> + *rb*2*φ*<sup>2</sup> =*q*0. Eq. (69) yields the following relation

$$F\_1(t) \approx F\_0(t) = k\_0 q\_0 + k\_0 e(t). \tag{71}$$

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

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

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 *cz* is assumed to be constant.

<sup>1</sup> + *rb*1*kz*(*t*) *rb*1*φ*<sup>1</sup> + *rb*2*φ*<sup>2</sup> + *e*(*t*) + *rb*1*cz rb*1*φ*˙ <sup>1</sup> + *rb*2*φ*˙ <sup>2</sup> + *e*˙(*t*) =*M*1(*t*), (66)

<sup>2</sup> + *rb*2*kz*(*t*) *rb*1*φ*<sup>1</sup> + *rb*2*φ*<sup>2</sup> + *e*(*t*) + *rb*2*cz rb*1*φ*˙ <sup>1</sup> + *rb*2*φ*˙ <sup>2</sup> + *e*˙(*t*) =*M*2(*t*). (67)

*<sup>i</sup>* (*i* = 1,2) are rotation angle, angular velocity, angular acceleration of the in‐

*q* =*rb*1*φ*<sup>1</sup> + *rb*2*φ*2. (68)

¨ <sup>+</sup> *kz*(*t*)*<sup>q</sup>* <sup>+</sup> *czq*˙ <sup>=</sup> *<sup>F</sup>* (*t*)−*kz*(*t*)*e*(*t*)−*cze*˙(*t*), (69)

+

*M*2(*t*)*rb*<sup>2</sup> *J*2

*F* (*t*)≈ *F*0(*t*)=*k*0*q*<sup>0</sup> + *k*0*e*(*t*). (71)

). (70)

put pinion and the output wheel respectively. *J* 1 and *J* 2 are the mass moments of inertia of the gears. *M* 1(*t*) and *M* 2(*t*) denote the external torques load applied on the system. *r <sup>b</sup>*1 and *r*

<sup>2</sup> *<sup>F</sup>* (*t*)=*mred* ( *<sup>M</sup>*1(*t*)*rb*<sup>1</sup>

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*˙ <sup>1</sup> =*ω*<sup>1</sup> =*const*, *φ*˙ <sup>2</sup> =*ω*<sup>2</sup> =*const*, *cz* =0, *kz*(*t*)=*k*0, the transmission error *q* is equal to the static tooth deflection under constant load *q*0 as *q* =*rb*1*φ*<sup>1</sup> + *rb*2*φ*<sup>2</sup> =*q*0. Eq. (69) yields the following

*J*1

The differential equations of motion for this system can be expressed in the form

*<sup>b</sup>*2 represent the base radii of the gears. By introducing the composite coordinate

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

*mred q*

*mred* <sup>=</sup> *<sup>J</sup>*1*J*<sup>2</sup> *J*1*rb*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>J</sup>*2*rb*<sup>1</sup>

*J*1*φ*¨

*J*2*φ*¨

, *φ*˙ *<sup>i</sup>* , *φ*¨

where *φ<sup>i</sup>*

where

relation

rigid disks connected by a spring-damper set along the line of contact.

318 Advances in Vibration Engineering and Structural Dynamics

$$m\_{red}\ddot{\vec{q}} + k\_z(t)q + c\_z\dot{q} - f\left(t\right) = 0,\tag{72}$$

where *f* (*t*)=*k*0*q*<sup>0</sup> − *kz*(*t*)−*k*<sup>0</sup> *e*(*t*)−*cze*˙(*t*).

In steady state motion of the gear system, the mesh stiffness *k <sup>z</sup>*(*t*) can be approximately rep‐ resented by a truncated Fourier series [33]

$$k\_z(t) = k\_0 + \sum\_{n=1}^{N} k\_n \cos(n\omega\_z t + \gamma\_n). \tag{73}$$

where *ωz* is the gear meshing angular frequency which is equal to the number of gear teeth times the shaft angular frequency and *N* is the number of terms of the series.

In general, the error components are no identical for each gear tooth and will produce dis‐ placement 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


$$e(t) = \sum\_{i=1}^{l} e\_i \cos(i\omega\_1 t + \alpha\_i). \tag{74}$$

**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 in‐ clude *J*1= 0.093 (kgm2 ), *J* <sup>2</sup> = 0.272 (kgm2 ) and nominal pinion speed of 1800 rpm (*f* <sup>1</sup> = 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 *q*<sup>0</sup> = 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 *<sup>ω</sup>*¯ <sup>0</sup> <sup>=</sup> *<sup>k</sup>*<sup>0</sup> / *mred* <sup>≈</sup>5462s-1, corresponding to *f* ¯ <sup>0</sup> <sup>=</sup>*<sup>ω</sup>*¯ <sup>0</sup> / <sup>2</sup>*<sup>π</sup>* <sup>≈</sup>869 (Hz). Based on the experimental work, the mean value of the Lehr damping ratio *ζ*¯ =0.024 is used for the dynamic model. The damping coefficient *cz* can then be determined by *cz* =2*<sup>ω</sup>*¯ <sup>0</sup>*ζ*¯*mred* .

frequency *f* 1 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

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**Figure 10.** Modelling result: frequency spectrum of d*q*/d*t* corresponding to (a) excitation function *e*(*t*) of Case 1 and

The experiment was done at an ordinary back-to-back test rig (Figure 11). The major param‐ eters 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 fre‐

generating sidebands.

**Figure 9.** . Modelling result: dynamic transmission error *q*(*t*).

b) excitation function *e*(*t*) with larger coefficients (Case 2).


**Table 4.** Fourier coefficients and phase angles of the mesh stiffness.


**Table 5.** Fourier coefficients and phase angles of excitation function *e*(*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 *f* 1 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.

**Figure 9.** . Modelling result: dynamic transmission error *q*(*t*).

means of a FEM software [29]. The static tooth deflection is estimated to be *q*<sup>0</sup> = 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 *<sup>ω</sup>*¯ <sup>0</sup> <sup>=</sup> *<sup>k</sup>*<sup>0</sup> / *mred* <sup>≈</sup>5462s-1,

the Lehr damping ratio *ζ*¯ =0.024 is used for the dynamic model. The damping coefficient *cz*

*n kn***(N/m) γ***n***(radian)**

 3.2267107 2.5581 1.3516107 -1.4421 8.1510106 -2.2588 3.5280106 0.9367 4.0280106 -0.8696 9.7100105 -2.0950 1.4245106 0.9309 1.5505106 0.2584 4.6450105 -1.2510 1.4158106 2.1636

0 8.1846108

**Table 4.** Fourier coefficients and phase angles of the mesh stiffness.

**(mm) α***<sup>i</sup>*

**Case 1 Case 2**

**Table 5.** Fourier coefficients and phase angles of excitation function *e*(*t*) .

**(rad)** *ei*

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

 0.0015 -0.049 0.010 1.0470 0.0035 -1.7661 0.003 -1.4521 0.0027 -0.7286 0.0018 0.5233 0.0011 -0.5763 0.0011 1.4570 0.0005 -0.7810 0.0009 -0.8622 0.0013 1.8172 0.0003 1.1966

<sup>0</sup> <sup>=</sup>*<sup>ω</sup>*¯ <sup>0</sup> / <sup>2</sup>*<sup>π</sup>* <sup>≈</sup>869 (Hz). Based on the experimental work, the mean value of

**(mm) α***<sup>i</sup>*

**(rad)**

corresponding to *f*

*i*

*ei*

¯

320 Advances in Vibration Engineering and Structural Dynamics

can then be determined by *cz* =2*<sup>ω</sup>*¯ <sup>0</sup>*ζ*¯*mred* .

**Figure 10.** Modelling result: frequency spectrum of d*q*/d*t* 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 param‐ eters 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 fre‐ quency *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 *φ*˙ <sup>1</sup> and *φ*˙ 2) in order to deter‐ mine experimentally the dynamic transmission error. The measurement was taken with two non-contacting transducers mounted in proximity to the shafts, positioned at the closest po‐ sition 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 fail‐ ure occurred on some teeth of the pinion.

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 dis‐ played in Figures 13 and 14, it can be observed that the vibration spectrum calculated by nu‐ merical methods (Figure 13) and the spectrum of the measured vibration signal (Figure 14)

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show the same sideband structures.

**Figure 13.** Calculating result: frequency spectrum of d*q*/d*t.*

**Figure 14.** Experimental result: zoomed frequency spectrum of d*q*/d*t* from Figure 12.

computer was used as in the previous example.

The calculations required a large number of time steps to ensure that the frequency resolu‐ tion 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

**Figure 11.** Gearbox test rig.

**Figure 12.** Experimental result: frequency spectrum of d*q*/d*t.*

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 dis‐ played in Figures 13 and 14, it can be observed that the vibration spectrum calculated by nu‐ merical methods (Figure 13) and the spectrum of the measured vibration signal (Figure 14) show the same sideband structures.

**Figure 13.** Calculating result: frequency spectrum of d*q*/d*t.*

quency *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 *φ*˙ <sup>1</sup> and *φ*˙ 2) in order to deter‐ mine experimentally the dynamic transmission error. The measurement was taken with two non-contacting transducers mounted in proximity to the shafts, positioned at the closest po‐ sition 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 fail‐

ure occurred on some teeth of the pinion.

322 Advances in Vibration Engineering and Structural Dynamics

**Figure 11.** Gearbox test rig.

**Figure 12.** Experimental result: frequency spectrum of d*q*/d*t.*

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

**Figure 14.** Experimental result: zoomed frequency spectrum of d*q*/d*t* from Figure 12.

The calculations required a large number of time steps to ensure that the frequency resolu‐ tion 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.

The dynamic model of this system shown in Figure 17 is used to investigate periodic vibra‐ tions 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-damp‐

The rotating components are modeled by two rotating disks with moments of inertia *I*<sup>0</sup> and *I*1. The cam mechanism has a nonlinear transmission function *U* (*φ*1) as a function of the ro‐ tating angle *φ*1 of the cam shaft, the driving torque from the motor *M*(*t*) and the external

When the angular velocity *Ω* of the driver input is assumed to be constant in the steady state

where *q*1 is the difference between rotating angles *φ*0 and *φ*1 due to the presence of elastic element *k*1 and damping element *c*1, resulted from the flexible transmission mechanism.

By the analogous way as in Section 3.1, we obtain the linear differential equations of vibra‐

and constant damping coefficients *ci*

Parametric Vibration Analysis of Transmission Mechanisms Using Numerical Methods

*φ*<sup>0</sup> =*Ωt*, (75)

*φ*<sup>1</sup> =*Ωt* + *q*<sup>1</sup> (76)

*M* (*Ωt*)*q***¨** + *C*(*Ωt*)*q***˙** + *K*(*Ωt*)*q* =*d*(*Ωt*) (77)

, *i* =1, 2.

325

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ing elements with time-invariant stiffness *ki*

**Figure 17.** Dynamic model of the transport manipulator.

tion for the system in the compact matrix form as

one leads to the following relation

load *F*(*t*) applied on the system.

**Figure 15.** Comparison of the computation time for the gear-pair model.

#### **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.

**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 vibra‐ tions 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-damp‐ ing elements with time-invariant stiffness *ki* and constant damping coefficients *ci* , *i* =1, 2. The rotating components are modeled by two rotating disks with moments of inertia *I*<sup>0</sup> and *I*1. The cam mechanism has a nonlinear transmission function *U* (*φ*1) as a function of the ro‐ tating angle *φ*1 of the cam shaft, the driving torque from the motor *M*(*t*) and the external load *F*(*t*) applied on the system.

**Figure 17.** Dynamic model of the transport manipulator.

When the angular velocity *Ω* of the driver input is assumed to be constant in the steady state

$$
\varphi\_0 \equiv \Omega t \,\tag{75}
$$

one leads to the following relation

**Figure 15.** Comparison of the computation time for the gear-pair model.

324 Advances in Vibration Engineering and Structural Dynamics

depicted in Figure 16.

**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

**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).

$$
\varphi\_1 = \Omega t + q\_1 \tag{76}
$$

where *q*1 is the difference between rotating angles *φ*0 and *φ*1 due to the presence of elastic element *k*1 and damping element *c*1, resulted from the flexible transmission mechanism.

By the analogous way as in Section 3.1, we obtain the linear differential equations of vibra‐ tion for the system in the compact matrix form as

$$\overset{\circ}{M}(\varOmega t)\overset{\circ}{q} + \prescript{\circ}{C}(\varOmega t)\dot{q} + \mathop{\rm K}(\varOmega t)q = \prescript{\circ}{d}(\varOmega t) \tag{77}$$

$$\begin{aligned} \mathbf{M}(\boldsymbol{\Omega}t) &= \begin{bmatrix} \boldsymbol{I}\_{1} + \boldsymbol{m}\_{2}\boldsymbol{\overline{\boldsymbol{\boldsymbol{L}}}}^{\boldsymbol{\prime}\prime} & \boldsymbol{m}\_{2}\boldsymbol{\overline{\boldsymbol{\boldsymbol{L}}}}^{\boldsymbol{\prime}} \\ \boldsymbol{m}\_{2}\boldsymbol{\overline{\boldsymbol{\boldsymbol{L}}}}^{\boldsymbol{\prime}} & \boldsymbol{m}\_{2} \end{bmatrix} \mathbf{C}(\boldsymbol{\Omega}t) = \begin{bmatrix} \boldsymbol{c}\_{1} + 2\boldsymbol{m}\_{2}\boldsymbol{\Omega}\,\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}}\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}} & \boldsymbol{0} \\ 2\boldsymbol{m}\_{2}\boldsymbol{\Omega}\,\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}} & \boldsymbol{c}\_{2} \end{bmatrix} \\ \mathbf{K}(\boldsymbol{\Omega}t) &= \begin{bmatrix} \boldsymbol{k}\_{1} + \boldsymbol{F}\,\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}} + \boldsymbol{m}\_{2}\boldsymbol{\Omega}\,\boldsymbol{\Omega}^{2}(\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}}\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}} + \overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}}) & \boldsymbol{0} \\ \boldsymbol{m}\_{2}\boldsymbol{\Omega}\,\boldsymbol{\Omega}^{2}\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}} & \boldsymbol{k}\_{2} \end{bmatrix} \boldsymbol{d} = \begin{bmatrix} -\boldsymbol{F}\,\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}} - \boldsymbol{m}\_{2}\boldsymbol{\Omega}\,\boldsymbol{\boldsymbol{2}}^{2}\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}}\overline{\boldsymbol{\boldsymbol{U}}}^{\boldsymbol{\prime}} \\ -\boldsymbol{F} - \boldsymbol{m}\_{2}\boldsymbol{\Omega}\,\boldsymbol{\$$

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

$$\text{Lul}^{'}(\varphi) = \sum\_{k=1}^{K} \left( a\_k \cos k\varphi + b\_k \sin k\varphi \right) \tag{78}$$

**Figure 19.** Calculating result of *q*2 for case 2, (a) time curve, (b) frequency spectrum.

appropriate initial conditions for the vector of variables q.

two cases since |*<sup>ρ</sup>* <sup>|</sup> max <1.

stationary periodic vibrations only.

The Fourier coefficients *ak* in 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

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

Calculating results of periodic vibrations of the mechanical adjustment unit, i.e. periodic sol‐ utions 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 rotat‐ ing frequency at Ω, 3Ω, 5Ω. These spectrums indicate that the considered system performs

The functions *<sup>U</sup>*¯′ , *U*¯′′ , *<sup>U</sup>*¯‴ in Eq. (77) can then be calculated using Eq. (78) for *<sup>φ</sup>* <sup>=</sup>*Ωt*.

The following parameters are used for numerical calculations: Rotating speed of the driver input n=50 (rpm) corresponding to *<sup>Ω</sup>* =5.236(1 /*s*), stiffness *k*<sup>1</sup> =7692 Nm; *k*<sup>2</sup> =10<sup>6</sup> N/m, damping coefficients *c*<sup>1</sup> =18.5 Nms; *c*<sup>2</sup> =2332 Ns/m, *I*<sup>1</sup> =1.11 kgm2 and *m*<sup>2</sup> =136 kg.

**Figure 18.** Calculating result of *q*2 for case 1, (a) time curve, (b) frequency spectrum.

**Figure 19.** Calculating result of *q*2 for case 2, (a) time curve, (b) frequency spectrum.

*M* (*Ωt*)=

*K*(*Ωt*)=

The functions *<sup>U</sup>*¯′

*<sup>I</sup>*<sup>1</sup> <sup>+</sup> *<sup>m</sup>*2*<sup>U</sup>*¯′<sup>2</sup>

*<sup>m</sup>*2*<sup>U</sup>*¯′ *<sup>m</sup>*<sup>2</sup>

*<sup>k</sup>*<sup>1</sup> <sup>+</sup> *FU*¯″ <sup>+</sup> *<sup>m</sup>*2*<sup>Ω</sup>* 2(*U*¯′

, *U*¯′′

*m*2*U*¯′

326 Advances in Vibration Engineering and Structural Dynamics

*<sup>m</sup>*2*<sup>Ω</sup>* <sup>2</sup>

, *C*(*Ωt*)=

In steady state motion of the cam mechanism, function *U* ′

*U* ′

damping coefficients *c*<sup>1</sup> =18.5 Nms; *c*<sup>2</sup> =2332 Ns/m, *I*<sup>1</sup> =1.11 kgm2

**Figure 18.** Calculating result of *q*2 for case 1, (a) time curve, (b) frequency spectrum.

*<sup>U</sup>*¯‴ <sup>+</sup> *<sup>U</sup>*¯″<sup>2</sup>

*<sup>U</sup>*¯‴ *<sup>k</sup>*<sup>2</sup>

(*φ*)=∑ *k*=1 *K*

*<sup>c</sup>*<sup>1</sup> <sup>+</sup> <sup>2</sup>*m*2*ΩU*¯′

) 0

*<sup>U</sup>*¯″ <sup>0</sup>

<sup>−</sup> *FU*¯′

, *<sup>U</sup>*¯‴ in Eq. (77) can then be calculated using Eq. (78) for *<sup>φ</sup>* <sup>=</sup>*Ωt*.

<sup>−</sup>*m*2*<sup>Ω</sup>* <sup>2</sup>

<sup>−</sup> *<sup>F</sup>* <sup>−</sup>*m*2*<sup>Ω</sup>* <sup>2</sup>

*U*¯′ *U*¯″

*<sup>U</sup>*¯″ , *<sup>q</sup>* <sup>=</sup>

(*ak* cos*kφ* + *bk* sin*kφ*) (78)

(*φ*) takes the form [18, 23]

and *m*<sup>2</sup> =136 kg.

*q*1 *q*2

<sup>2</sup>*m*2*ΩU*¯″ *<sup>c</sup>*<sup>2</sup>

, *d* =

The following parameters are used for numerical calculations: Rotating speed of the driver input n=50 (rpm) corresponding to *<sup>Ω</sup>* =5.236(1 /*s*), stiffness *k*<sup>1</sup> =7692 Nm; *k*<sup>2</sup> =10<sup>6</sup> N/m,

> The Fourier coefficients *ak* in 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 |*<sup>ρ</sup>* <sup>|</sup> max of the solutions of the characteristic equation, according to Eq. (10), is now calculated. The obtained values for both cases are |*<sup>ρ</sup>* <sup>|</sup> max =0.001992 (case 1) and <sup>|</sup>*<sup>ρ</sup>* <sup>|</sup> max =0.001623 (case 2). It can be concluded that the system is dynamically stable for both two cases since |*<sup>ρ</sup>* <sup>|</sup> max <1.

> Calculating results of periodic vibrations of the mechanical adjustment unit, i.e. periodic sol‐ utions 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 rotat‐ ing frequency at Ω, 3Ω, 5Ω. These spectrums indicate that the considered system performs stationary periodic vibrations only.

**Acknowledgements**

**Author details**

Nguyen Van Khang\*

**References**

B.G. Teubner.

CRS Press.

er.

bridge University Press.

John Wiley & Sons.

Stuttgart, Teubner.

tial equations. Berlin, Springer.

tion). New York, Oxford University Press.

for Science and Technology Development (NAFOSTED).

and Nguyen Phong Dien

\*Address all correspondence to: nvankhang@mail.hut.edu.vn

*Journal of Engineering Mechanics; Division*, 85-67.

This study was completed with the financial support by the Vietnam National Foundation

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329

Department of Applied Mechanics, Hanoi University of Science and Technology, Vietnam

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**Figure 20.** 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 experi‐ mental result than the results obtained by the WKB-method and the kinesto-static calculation.
