Patent Publication Number: US-7916372-B2

Title: Movable body apparatus and optical deflector using the movable body apparatus

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to techniques of a movable body apparatus including at least a movable body which is reciprocally and rotatably supported. More particularly, the present invention relates to a movable body apparatus, such as a resonance-type movable body apparatus, and an optical deflector using this. This optical deflector can be preferably used in optical instruments, such as image forming apparatuses like a scanning-type display, a laser beam printer and a digital copying machine. 
     2. Description of the Related Art 
     In recent years, an optical deflector for deflecting and scanning a light beam is used in optical disc apparatuses, laser beam printers and the like. Further, there have been proposed optical deflectors with a minute mirror produced using micro-machining techniques and capable of vibration in a resonance manner. 
     Such a resonance-type optical deflector has advantages as follows. Compared with an optical deflector using a rotary polygonal mirror, the size can be greatly reduced. The consumption electrical power also can be reduced. There theoretically exists no problem of so-called face tangle of a reflecting surface. In particular, with such an optical deflector formed of a Si single crystal capable of being fabricated by a semiconductor processing method, no metal fatigue exists theoretically, and the endurance property is typically excellent. 
     However, in resonance-type optical deflectors, the deflection angle (angular displacement) of a mirror changes in a sine-wave fashion. Accordingly, the angular velocity of the mirror varies. Japanese Patent Application Laid-Open No. 2005-208578 A (first Japanese reference, corresponding to U.S. Pat. Nos. 7,271,943 and 7,388,702, and US2008204843) discloses a method for correcting such property of a varying angular velocity. The deflection angle of a mirror has a predetermined relationship with the scan angle of a light beam deflected by the mirror, and hence these angles can be equivalently used. In this specification, the deflection angle (angular displacement) and the scan angle are used as a term having the same meaning. 
     The first Japanese reference discloses a micro-movable body apparatus in which a vibratory system with plural torsion springs and plural movable bodies has plural discrete characteristic vibratory modes. In this micro-movable body apparatus, plural discrete characteristic vibratory modes include a fundamental vibratory mode with a fundamental frequency and an even number-fold vibratory mode with a frequency equal to an approximately even number-fold of the fundamental frequency. 
     In this micro-movable body apparatus, a saw-tooth wave drive with an approximately equi-angular velocity range is achieved by vibrating the micro-movable body in those vibratory modes. The saw-tooth wave drive is illustrated in  FIG. 22 . In this drive, an angular displacement time of one way in a round trip motion of the movable body differs from that of the other way in the round trip motion within each period of the angular displacement. When the light beam deflected by the minute mirror under vibration of the saw-tooth wave drive is corrected by a correcting system or the like, approximately equi-velocity of a light spot formed on a scan surface can be attained without any change in the diameter of the light spot. 
     Meanwhile, the resonance-type optical deflector has the property that the resonance frequency changes due to a change in the ambient condition such as temperature. Japanese Patent Application Laid-Open No. 1995 (Heisei 7)-181415 A (second Japanese reference) discloses techniques of self-excited vibration as follows. An output signal of a detector for detecting the vibration of a movable body is fed back to a vibration input portion to control the driving frequency of the movable body. Thus, the movable body is always vibrated at its resonance frequency in response to a change in temperature. 
     When techniques of the second Japanese reference are applied to a system of plural torsion springs and plural movable bodies as disclosed in the first Japanese reference, the following disadvantage occurs. In techniques of the second Japanese reference, a delay phase difference between a target drive signal and the vibration of the torsion spring has a fixed value. In a case where plural factors exist for the phase delay between the drive signal and the vibration of the torsion spring, it is not easy to obtain an accurate delay phase difference. 
     Further, in techniques of the second Japanese reference, driving is performed in a single vibratory mode (bending deformation mode or torsional deformation mode). Therefore, when the driving of plural torsion springs is performed at a resonance frequency in the same kinds of plural vibratory modes about a common axis, the following disadvantage occurs. 
     For example, a vibratory system  200  as illustrated in  FIG. 2  includes movable bodies  201  and  202 , a torsion spring  211  for coupling these movable bodies, and a torsion spring  212  for coupling the movable body  202  to a support portion  221 . In order that the movable body is driven as illustrated in  FIG. 22 , the driving needs to be performed by a combined wave of a fundamental driving wave in a fundamental vibratory mode at a frequency near a resonance frequency and an integer-fold driving wave at a double frequency, as illustrated in  FIG. 19 . When two movable bodies are driven by the combined wave having two frequency components in the same kinds of vibratory modes, it is not easy to control two frequency components of the angular displacement unless phases of frequency components of the drive signal are controlled. 
     Furthermore, in techniques of the second Japanese reference, it is not easy to drive the movable body at a frequency intentionally deviated from the resonance frequency. 
     SUMMARY OF THE INVENTION 
     According to one aspect, the present invention provides a movable body apparatus which includes a vibratory system, a vibration detecting portion, a driving portion, and a controlling portion. The vibratory system has a resonance frequency and a movable body capable of being reciprocally and rotatably vibrated. The vibration detecting portion is configured to detect a vibration condition of the movable body. The driving portion is configured to drive the vibratory system with a drive signal. The controlling portion is configured to regulate the drive signal supplied to the driving portion. 
     The controlling portion stores, as a target delay phase difference, a delay phase difference between a drive phase of the drive signal and a vibration phase of the vibratory system obtained from a detection result of the vibration detecting portion, both the drive phase and the vibration phase being obtained at the time when the vibratory system is vibrated at a predetermined frequency. The controlling portion regulates a driving frequency of the drive signal so that the delay phase difference between the drive phase of the drive signal and the vibration phase of the vibratory system obtained from detection result of the vibration detecting portion, both measured during driving of the vibratory system, is caused to be approximately coincident with the target delay phase difference. 
     According to another aspect, the present invention provides an optical deflector which includes the above-described movable body apparatus, and a reflective mirror provided on the movable body to reflect and deflect a light beam from a light source. The vibration detecting portion includes a light receiving device arranged to detect the deflected light beam at a predetermined deflection angle, and the vibration condition of the movable body is detected based on a time interval of light beam detection by the light receiving device. 
     According to yet another aspect, the present invention provides an optical instrument which includes the above-described optical deflector, and an irradiation target object. The optical deflector deflects the light beam from the light source, and directs at least a portion of the light beam to the irradiation target object. 
     According to the present invention, in a vibratory system with plural movable bodies as well as a vibratory system with a single movable body, the driving frequency for achieving an efficient driving can be determined even if the resonance frequency changes due to a change in the ambient condition such as temperature. 
     Further features of the present invention will become apparent from the following description of exemplary embodiments with reference to the attached drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram illustrating the construction of an embodiment of an optical deflector using a movable body apparatus according to the present invention. 
         FIG. 2  is a view illustrating an optical deflector including two movable bodies. 
         FIG. 3  is a view illustrating the deflection angle of an optical deflector. 
         FIG. 4A  is a plan view illustrating an example of a vibratory system in an optical deflector. 
         FIG. 4B  is a cross-sectional view illustrating an example of a driving portion in the optical deflector. 
         FIG. 5  is a graph illustrating a change with time in the deflection angle in an optical deflector. 
         FIG. 6A  is a block diagram illustrating the construction of an NCO acting as a waveform generator. 
         FIG. 6B  is a graph showing output waveforms of the NCO illustrated in  FIG. 6A . 
         FIG. 7A  is a graph showing the driving frequency-sensitivity characteristic. 
         FIG. 7B  is a graph showing the driving frequency-amplitude characteristic. 
         FIG. 7C  is a graph showing driving the driving frequency-delay phase characteristic. 
         FIG. 8A  is a graph showing a change in the driving frequency-amplitude characteristic due to a change in temperature. 
         FIG. 8B  is a graph showing a change in the driving frequency-delay phase characteristic due to a change in temperature. 
         FIG. 9A  is a graph showing a manner of changing the driving frequency in response to a change in resonance characteristic. 
         FIG. 9B  is a graph showing a manner of changing the driving frequency in response to a change in resonance characteristic. 
         FIG. 10A  is a graph showing the driving frequency-amplitude characteristic of a system with two movable bodies in a case of Δω=0. 
         FIG. 10B  is a graph showing the driving frequency-delay phase characteristic of the system with two movable bodies in a case of Δω=0. 
         FIG. 11A  is a graph showing the driving frequency-amplitude characteristic of a system with two movable bodies in a case of Δω&gt;0. 
         FIG. 11B  is a graph showing the driving frequency-delay phase characteristic of the system with two movable bodies in a case of Δω&gt;0. 
         FIG. 12A  is a graph showing the driving frequency-amplitude characteristic of a system with two movable bodies in a case of Δω&lt;0. 
         FIG. 12B  is a graph showing the driving frequency-delay phase characteristic of the system with two movable bodies in a case of Δω&lt;0. 
         FIG. 13A  is a side view illustrating an image forming apparatus using a movable body apparatus as an optical deflector. 
         FIG. 13B  is a plan view illustrating the image forming apparatus in  FIG. 13A . 
         FIG. 14  is a flow chart of a first exemplary embodiment as illustrated in  FIG. 1 . 
         FIG. 15  is a graph showing a manner of changing the driving frequency in a second exemplary embodiment using a threshold. 
         FIG. 16A  is a view illustrating image and non-image describing regions in a third exemplary embodiment. 
         FIG. 16B  is a view illustrating image and non-image describing regions in the third exemplary embodiment. 
         FIG. 17  is a block diagram illustrating a controlling portion of the third exemplary embodiment. 
         FIG. 18  is a flow chart of the third exemplary embodiment in  FIG. 17 . 
         FIG. 19  is a graph showing the relationship between waveforms of components of drive signal and angular displacement. 
         FIG. 20  is a graph showing the relationship between waveforms of drive signal and angular displacement in a fourth exemplary embodiment of a movable body apparatus with a single movable body. 
         FIG. 21  is a flow chart of the fourth exemplary embodiment in  FIG. 20 . 
         FIG. 22  is a graph showing a saw-tooth wave of the angular displacement of a movable body. 
     
    
    
     DESCRIPTION OF THE EMBODIMENTS 
     Embodiments of the present invention will hereinafter be described. 
     A fundamental embodiment of a movable body apparatus according to the present invention includes a vibratory system, a vibration detecting portion for detecting the vibration condition of a movable body in the vibratory system, a driving portion for driving the vibratory system by a drive signal, and a controlling portion for controlling or regulating the drive signal supplied to the driving portion. The vibratory system has a resonance frequency, and includes at least a movable body that is reciprocally and rotatably supported. 
     The controlling portion stores, as a target delay phase difference, the delay phase difference between a drive phase of the drive signal and a vibration phase of the vibratory system obtained through detection result by the vibration detecting portion, both obtained at the time when the vibratory system is vibrated at a predetermined frequency. Further, the controlling portion regulates a driving frequency of the drive signal so that the delay phase difference measured during driving is caused to be approximately coincident with the target delay phase difference. 
     When the vibratory system has a single movable body, the vibratory system has a single resonance frequency. The drive signal has a signal component with a frequency, and the vibration of the movable body has a single frequency component. Therefore, there is no need to regulate a phase difference between plural frequency components. 
     Thus, it is easy to acquire the delay phase difference between a single drive phase of the drive signal and the vibration phase of a single movable body obtained through the detection result by the vibration detecting portion. In this case, the delay phase difference at the time of vibration at a predetermined frequency is beforehand measured and stored as the target delay phase difference. A single driving frequency of the drive signal only needs to be regulated so that the delay phase difference measured during driving is caused to be approximately coincident with the target delay phase difference. 
     In a case where the vibratory system includes plural movable bodies, the vibratory system has plural resonance frequencies. The drive signal needs to have signal components with plural frequencies, and the vibration of the movable body has plural frequency components. Hence, there is a necessity that the controlling portion regulates a phase difference between plural frequency components of the drive signal so that a phase difference between plural frequency components involved in the vibration of the movable body reaches a predetermined value. Here, the controlling portion changes frequencies of plural signal components while maintaining an integer ratio between these frequencies. 
     The controlling portion also measures the relationship between frequencies of signal components of the drive signal corresponding to frequency components of the vibration of the movable body and amplitudes of the frequency components of the vibration, and determines plural driving frequencies based on the measurement result. Further, the controlling portion beforehand measures the delay phase difference (the delay phase difference between the drive phase of the signal component of the drive signal corresponding to any one of plural frequency components of the vibration of the movable body and the vibration phase thereof) at the time of vibration at a predetermined driving frequency, and stores it as the target delay phase difference. The controlling portion regulates plural driving frequencies of the drive signal so that the above delay phase difference measured during driving is caused to be approximately coincident with the target delay phase difference. In this specification, “approximately coincident” means acquisition of a phase difference closest to the target delay phase difference achieved by control of the driving portion with a possible frequency changing precision. 
     According to the above-described construction, in a vibratory system with plural movable bodies as well as a vibratory system with a single movable body, the driving frequency for achieving efficient driving can be determined even if the resonance frequency varies. 
     A specific first exemplary embodiment will be described.  FIG. 1  illustrates the construction of the first exemplary embodiment relating to an optical deflector using a movable body apparatus of the present invention.  FIG. 2  illustrates the configuration of a vibratory system of the optical deflector. 
     As illustrated in  FIG. 2 , a vibratory system  200  includes first and second movable bodies  201  and  202  that are reciprocally and rotatably supported. The movable bodies  201  and  202  are serially coupled by a torsion spring  211 , and the second movable body  202  is coupled to a support portion  221  by a torsion spring  212 . 
     A driving portion  220  applies to the movable body a driving force for simultaneously exciting plural characteristic vibratory modes (here, plural vibratory modes of torsional deformation modes) by an electromagnetic, electrostatic, or piezoelectric method. For example, the electromagnetic driving portion is comprised of a coil and a permanent magnet. The movable body  201  has a reflective mirror  230  on its surface, and a light beam  232  from a light source  231  is reflected and deflected thereby. A scanning light beam  233  passes a vibration detecting portion or first and second light receiving devices  240  and  260  twice within a period. A controlling portion  250  generates drive signals supplied to the driving portion  220 , by using times at which the scanning light beam  233  passes the light receiving devices  240  and  260 . 
     The relationship between the scanning light beam  233  and the light receiving devices  240  and  260  will be described.  FIG. 3  illustrates the deflection angle of the light beam deflected by the movable body  201  in the vibratory system  200 . The movable body  201  in the vibratory system  200  is reciprocally and rotatably moved with the angular displacement (deflection angle) of θmax. The first and second light receiving devices  240  and  260  are arranged at locations where the scanning light beam  233  deflected by an angle θ 1  smaller than θmax passes. When the maximum deflection angle θmax of the scanning light beam  233  is sufficiently larger than θ 1 , the scanning light beam  233  passes each of the light receiving devices  240  and  260  twice within a period, as illustrated in  FIG. 19 . 
       FIG. 4  shows an example of a portion of the vibratory system  200  and the driving portion  220 .  FIG. 4A  is a plan view of an optical deflector of the vibratory system. A plate member  400  is fabricated by etching a silicon wafer, for example. A planar movable body  401  is supported by a torsion spring  411 . An optical reflective film  431  is deposited on an upper surface of the movable body  401 . A movable body  402  supported by the torsion spring  411  is coupled to a support frame by a torsion spring  412 . A system of the movable bodies  401  and  402 , and the torsion springs  411  and  412  have two vibratory modes. The vibratory system  200  is constructed so that resonance frequencies of two vibratory modes include a fundamental resonance frequency and an approximately integer-fold (here approximately twice) frequency of the fundamental resonance frequency. That is, a ratio between the fundamental resonance frequency and integer-fold wave resonance frequency is an approximately integer ratio. In this specification, “approximately integer-fold” covers a range between 0.98n of the fundamental frequency and 1.02n of the fundamental frequency (n is an integer). 
       FIG. 4B  is a schematic view showing the driving portion  220  for the optical deflector.  FIG. 4B  illustrates a cross section taken along a line  490 . A magnet  441  is fixed on a lower surface of a movable body  402 . The plate member  400  is fixed to a yoke  444  formed of a material having a large permeability. A core  443  formed of a material having a large permeability is arranged on a portion of the yoke  444  facing the permanent magnet  441 . A coil  442  is wound around the core  443 . The magnet  441 , coil  442 , core  443  and yoke  444  constitute the driving portion of an electromagnetic actuator. When a current flow is caused in the coil  442 , a torque acts on the permanent magnet  441 . Thus, the movable body  402  is driven. 
     An ordinary driving of the vibratory system  200  by the driving portion  220  is performed in the following manner. The deflection angle θ of the optical deflector is represented by a formula (1) which is a function of time t.
 
θ( t )= A 1 sin(ω1 t )+ A 2 sin(ω2 t +φ)  (1)
 
where A 1  and ω 1  are amplitude and angular frequency of a first vibratory motion in one vibratory mode, A 2  and ω 2  are amplitude and angular frequency of a second vibratory motion in the other vibratory mode, and φ is the relative phase difference between two frequency components.
 
       FIG. 19  shows time waveforms of frequency components of the drive signal and the angular displacement (deflection angle). With respect to the drive signal, the phase difference between signal components of fundamental driving wave and integer-fold driving wave is set at φd so that φ is zero (i.e., φ=0) in the formula (1) of the angular displacement. In other words, the phase difference between plural frequency components involved in the vibration of the first movable body should be a predetermined value, and this is set at zero here. 
     The zero-crossing point of the angular displacement can be approximately calculated as an intermediate point between points of time at which the scanning light beam passes the first and second light receiving devices  240  and  260 . However, in a case of φ≠0, since the zero-crossing point of each frequency component of the angular displacement differs from that of the combined wave thereof, the calculation becomes complicated and an accurate value cannot be readily acquired. Accordingly, the controlling portion  250  regulates the drive signal so that φ=0 is achieved, and the vibratory system  200  is thus driven. 
     When intervals of time between the zero-crossing point of each frequency signal component of the drive signal and the zero-crossing point of the angular displacement are represented in the term of each frequency, the delay phase difference between the drive phase of the drive signal and the vibration phase of the vibratory system can be obtained. As illustrated in  FIG. 19 , the delay phase difference between the drive phase of the fundamental driving wave and the vibration phase is φ 1 , and the delay phase difference between the drive phase of the integer-fold driving wave and the vibration phase is φ 2 . 
     The controlling portion  250  for supplying the drive signal to the driving portion  220  will be described with reference to  FIG. 1 . 
     In the construction of  FIG. 1 , a controller  100  in the controlling portion  250  sets the angular frequency ω 1  of the first vibratory motion in a waveform generator  20 . The waveform generator  20  outputs sine-waves with angular frequencies ω 1  and 2×ω 1 . The phase difference φd between sine-waves with angular frequencies ω 1  and 2×ω 1  is calculated by a calculator  30 , and the calculated result is supplied through an integrator  40 . The two sine-waves thus generated are respectively multiplied by amplitudes A 1  and A 2  by a multiplier, and a combined wave generated through addition by an adder is supplied to the driving portion  220 . 
     In this specification, the drive signal for generating the deflection angle θ of the optical deflector is represented using factors corresponding to A 1 , ω 1 , A 2 , ω 2  and φ, and these factors are indicated by A 1 , ω 1 , A 2 , ω 2  and φd. It will be apparent from the context that common signs A 1 , ω 1 , A 2  and ω 2  are relevant to the deflection angle θ or the drive signal. 
     As illustrated in  FIG. 2 , the drive signal in the form of the combined wave from the controlling portion  250  is supplied to the driving portion  220  to apply the driving force to the vibratory system  200 . The movable body  201  is thus vibrated. The light beam  232  is deflected and scanned by the vibration of the movable body  201  with the mirror  230 . The scanning light  233  is received by the light receiving devices  240  and  260 . Reception times are t 1 , t 2 , t 3  and t 4 , as illustrated in  FIG. 5 . The controlling portion  250  takes differences (intervals between times of detection of the light beam) from t 1 , t 2 , t 3  and t 4 , and t 2 −t 1 , t 3 −t 2  and t 4 −t 3  are set in portions  121 ,  122  and  123  of a time measuring portion  120  acting as the vibration detecting portion, respectively. Differences between the calculated intervals of time set in portions  121 ,  122  and  123  and target times  110 ,  111  and  112  for achieving a desired angular displacement are acquired, and the calculator  30  converts these differences into operation amounts Δφd, ΔA 1 , ΔA 2  of the drive signal. 
     An exemplary calculating method by the calculator  30  will be described. The following coefficients and matrix M are beforehand acquired. The coefficients represent changes in intervals t 2 −t 1 , t 3 −t 1  and t 4 −t 1  relevant to passing times of the scanning light beam  233  through the first and second light receiving devices  240  and  260 , which occur when a control parameter X of A 1 , A 2  or φd minutely changes from a target value. Those are represented by the following formulae (2) and (3). 
     
       
         
           
             
               
                 
                   
                     
                       
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     Accordingly, operation amounts ΔA 1 , ΔA 2  and Δφd of amplitude and phase of each frequency component of the drive signal can be obtained from time differences Δt 2 , Δt 3  and Δt 4  between intervals t 2 −t 1 , t 3 −t 1  and t 4 −t 1  and target times t 20 −t 10 , t 30 −t 10  and t 40 −t 10 , using the following formula (4). Thus, the controlling portion  250  generates the drive signal supplied to the driving portion  220 . 
     
       
         
           
             
               
                 
                   
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     The construction and function of the controlling portion  250  will be described. In the controller  100 , a frequency for starting the drive is stored in a portion  101 , and at a drive start time the drive starting frequency in the portion  101  is set in the waveform generator  20 . The driving is thus started. Further, in the controller  100 , delay phase differences are stored in a portion  102 . The delay phase differences are differences between the drive phases of two signal components of the drive signal from the waveform generator  20  and the vibration phase of the angular displacement of the movable body  201  obtained from signals from the time measuring portion  120  (see  FIG. 19 ). 
     The waveform generator  20  can be comprised of an NCO (Numerical-Controlled-Oscillator), for example. An example of an NCO  60  is illustrated in  FIG. 6A . Addition of a digital input and a signal one sample prior thereto generated by a sampling delay device  62  is executed by an adder  61 , and the added result is input into a sine-wave table  63  as an address. A digital sine-wave component is derived from the sine-wave table  63 . In this construction, when a digital signal having a given level continues to be input, a signal with a given positive inclination according to the input level appears as an output of the adder  61 , as illustrated in  FIG. 6B . Provided that the digital input is reset to zero upon reaching a predetermined maximum value, it is possible to obtain a saw-tooth wave  64  with a period (frequency) according to the input level. Therefore, a sine-wave component  65  can be derived by inputting the saw-tooth wave  64  into the sine-wave table  63 . The frequency of the sine-wave component  65  can be changed by changing the input level. 
     In the controlling portion  250 , the following operation is also performed.  FIG. 7A  shows sensitivity near the resonance frequency of the movable body.  FIG. 7B  shows amplitude of the drive signal at the time of control.  FIG. 7C  shows frequency characteristic of the delay phase difference between the drive phase and the vibration phase. Here, the relationship between each signal component of the drive signal and the frequency component of the angular displacement corresponding thereto is illustrated. The sensitivity becomes maximum at the resonance frequency. Hence, in order that the target angular displacement is maintained by controlling, an intensity of the drive signal needs to be changed in accordance with the amount of deviation between the driving frequency and the resonance frequency. 
     The phase difference between the drive phase and the vibration phase decreases (i.e., the delay decreases) when the driving frequency is smaller than the resonance frequency, and increases (i.e., the delay increases) when the driving frequency is larger than the resonance frequency. Further, a changing rate of the delay phase difference is maximum at the resonance frequency. 
     Each resonance frequency can be detected as follows. The controlling portion  250  supplies the drive signal in the form of a single sine-wave to the driving portion  220  for driving the vibratory system  200  while sweeping the driving frequency of the drive signal near the resonance frequency, and measures the driving frequency-amplitude characteristic to detect the resonance frequency. Simultaneously, the controlling portion  250  detects the drive amplitude of the drive signal (in this case, the amplitude of the angular displacement is maintained at a constant value), or the amplitude of the vibration of the movable body (in this case, the drive amplitude is maintained at a constant value) to measure the driving frequency-amplitude characteristic. In the former case, the resonance frequency is a driving frequency at which the drive amplitude is minimum. In the latter case, the resonance frequency is a driving frequency at which the amplitude of the angular displacement is maximum. 
     Further, the resonance frequency can also be obtained based on attenuation of the vibration of the movable body occurring after the driving of the movable body is stopped. In this case, the resonance frequency can be measured from a counter electromotive force appearing in the driving coil, for example. 
       FIG. 8A  shows a change in the driving frequency-amplitude characteristic at the time when temperature increases,  FIG. 8B  shows a change in the driving frequency-phase characteristic at the time when temperature increases. It can be seen therefrom that the resonance frequency of the movable body decreases as temperature increases. In a case where the driving is performed at the resonance frequency prior to a change in temperature, the amplitude of the drive signal and the delay phase difference between the drive phase and the vibration phase increase due to a change in temperature. In order that the driving is always performed at the resonance frequency irrespective of a change in temperature, the delay phase difference between the drive phase and the vibration phase should be maintained even when temperature changes, as illustrated in  FIGS. 9A and 9B . 
     The operation for maintaining the delay phase difference will be described. In this operation, when the driving frequency is closer to the resonance frequency of the first vibratory mode, a precision of following the resonance frequency increases more when the target delay phase difference is set at the phase difference φ 1  between the fundamental driving wave phase and the vibration phase (see  FIG. 19 ) than when the target delay phase difference is set at the phase difference φ 2  between the integer-fold driving wave phase and the vibration phase. Conversely, when the driving frequency is closer to the resonance frequency of the second vibratory mode, the following precision increases more when the target delay phase difference is set at φ 2  than when the target delay phase difference is set at φ 1 . 
     Therefore, the delay phase difference is preferably the delay phase difference between the vibration phase of the movable body and the drive phase of a signal component out of the fundamental wave signal component and the integer-fold wave signal component, the frequency of which is closer to the resonance frequency than that of the other. This exemplary embodiment is constructed based on this principle. 
     As described above, the system of two vibrators as illustrated in  FIG. 2  has two vibratory modes, and the ratio between resonance frequencies of the vibratory modes is regulated to be approximately 1:2. However, the ratio between resonance frequencies actually deviates from 1:2. Where the fundamental resonance frequency is ω 0   1  and the integer-fold resonance frequency is ω 0   2 , a deviation difference Δω is defined by equation (5).
 
Δω=ω 0 2−(2×ω 0 1)  (5)
 
       FIGS. 10A and 10B  respectively show the drive signal amplitude-frequency characteristic and the phase difference-frequency characteristic in a device with Δω=0. Here, where the driving frequency of the fundamental wave component in the drive signal is ω d   1  and the driving frequency of the integer-fold wave component in the drive signal is ω d   2 , the relationship of equation (6) is always maintained.
 
ω d 2=2×ω d 1  (6)
 
     Accordingly, the driving at the resonance frequencies can be attained by establishing the following equations (7-a) and (7-b).
 
ω d 1=ω 0 1  (7-a)
 
ω d 2=ω 0 2  (7-b)
 
       FIGS. 11A and 11B  show the drive signal amplitude-frequency characteristic and the phase difference-frequency characteristic in a device with Δω&gt;0. Here, when the driving frequency of the fundamental wave component in the drive signal is caused to coincide with the fundamental resonance frequency, the driving frequency ω d   1  of the fundamental wave component in the drive signal and the driving frequency ω d   2  of the integer-fold wave component in the drive signal are represented by equations (8-a) and (8-b).
 
ω d 1=ω 0 1  (8-a)
 
ω d 2=2×ω 0 1=ω 0 2−Δω  (8-b)
 
     Then, the delay phase difference between the integer-fold driving wave phase and the vibration phase becomes smaller than when the driving frequency ω d   2  is set at the resonance frequency ω 0   2 . Further, the changing rate of the delay phase difference φ 1  on the side of the fundamental frequency is larger. Therefore, when the driving frequencies are changed by causing the driving frequency of the fundamental wave component in the drive signal to coincide with the fundamental resonance frequency as shown in formulae (8-a) and (8-b), the precision becomes higher when two driving frequencies are changed with the delay phase difference value φ 1  on the side of the fundamental frequency being the target delay phase difference. 
     Conversely, when the following equations (9-a) and (9-b) are established, the delay phase difference between the integer-fold driving wave phase and the vibration phase becomes larger than when the driving is performed by setting the driving frequency ω d   1  of the fundamental wave component at the resonance frequency ω 0   1 .
 
ω d 1=ω 0 2/2=ω 0 1+(Δω/2)  (9-a)
 
ω d 2=ω 0 2  (9-b)
 
     Further, the changing rate of the delay phase difference ω 2  on the side of the integer-fold frequency is larger. Therefore, in a case where the driving frequencies are changed by causing the driving frequency of the integer-fold wave component in the drive signal to coincide with the resonance frequency of the integer-fold wave as shown in formulae (9-a) and (9-b), the precision becomes higher when two driving frequencies are changed with the delay phase difference value φ 2  on the side of the integer-fold frequency being the target delay phase difference. 
     Further, when the driving is performed with the driving frequencies being in a range between two resonance frequencies, a way of changing two driving frequencies with the delay phase difference value φ 1  or φ 2  being the target delay phase difference can be determined based on the fact that to which of the fundamental and integer-fold resonance frequencies the driving frequencies are closer. 
       FIGS. 12A and 12B  show the drive signal amplitude-frequency characteristic and the phase difference-frequency characteristic in a device with Δω&lt;0, respectively. Also in this case, similar to the case of Δω&gt;0, which of the delay phase difference values φ 1  and φ 2  should be the target delay phase difference can be determined based on the fact that with which of the fundamental and integer-fold resonance frequencies ω 0   1  and ω 0   2  the driving frequency is caused to coincide. 
     Based on the above description of the construction and function, a specific frequency following manner will be described with reference to the flow chart of  FIG. 14 . The description will be made of an example in which the movable body apparatus is used in an image forming apparatus described below. 
     Initially, an electrical power source of the image forming apparatus is turned on (S 101 ). The controller  100  sets the starting driving frequency in the signal generator  20 , and sets the initial drive amplitude A 1  in the integrator  40  to execute the driving by the drive signal including the fundamental wave (S 102 ). As the starting driving frequency, an average value of resonance frequencies or the resonance frequency at the time of the last driving stop is used, for example. An initial value large enough to direct the scanning light beam  233  to the light receiving devices  240  and  260  is used as A 1 . 
     Upon application of the drive signal to the driving portion  220 , the movable body  201  is vibrated. The vibration is continued until the vibration amplitude of the movable body  201  increases and the signal enters the light receiving devices  240  and  260  (S 103 ). 
     Then, the drive signal including the integer-fold wave component is applied (S 104 ). The controller  100  sets the vibration amplitude A 2  of the initial integer-fold wave component and the initial phase difference φd in the integrator  40  (see  FIG. 19 ). After the integer-fold wave is applied, the controller  100  starts such a control that a desired angular displacement is obtained (S 105 ). This control is executed by the calculation using the above matrix. When the angular displacement of the movable body is converged to the desired angular displacement under such control, one of the driving frequencies is swept near a predicted resonance frequency. Thus, two driving frequencies are changed with the relationship of 1:2 being maintained (S 106 ). The drive amplitude at that time is measured (S 107 ). Until the measurement of the driving frequency-amplitude characteristic is completed, changing of the driving frequencies and measurement of the drive amplitude are repeated (S 108 ). Such operation is performed for the driving frequency corresponding to each of the two resonance frequencies. 
     Then, the driving frequency is determined based on the thus-measured driving frequency-amplitude characteristic (S 109 ). Here, description will be made of a case where the driving frequency ω d   2  is caused to coincide with the resonance frequency ω 0   2  of the integer-fold wave mode. In the driving at the driving frequency, the delay phase difference φ 2  between the integer-fold driving wave phase and the vibration phase is measured (S 110 ) The delay phase difference is stored in the storing portion  102  of the controller  100  as the target delay phase difference (S 111 ). As described above, it is also possible to perform the driving in which the driving frequency ω d   1  is caused to coincide with the fundamental resonance frequency ω 0   1 , or the driving in which driving frequencies are set in a range between two resonance frequencies. 
     After completion of the above preparation, actual printing is started (S 112 ). After the start, judgment if the printing is continued or finished is executed (S 113 ). When the printing (driving) is to be continued, the delay phase difference φ 2  between the integer-fold driving wave phase and the vibration phase during the driving is measured (S 114 ). The measured phase difference is compared with data (the target delay phase difference) stored in the storing portion  102  (S 115 ). Based on the comparison result, printing is continued when no difference is present. When the difference exists, two driving frequencies are changed with the relationship of 1:2 being maintained so that the measured delay phase difference φ 2  is caused to be approximately coincident with the target delay phase difference stored in the storing portion  102  (S 116 ). Further, when the printing is required to be finished in the conditional step (S 113 ), the printing is finished and the stand-by is started (S 117 ). 
     The above printing process in the image forming apparatus will be described with reference to  FIGS. 13A and 13B . 
     Upon request of the printing process, the optical deflector with the vibratory system  200  is brought into a steady drive condition through the above steps, and emits the scanning light beam  233 . The scanning light beam  233  is scanned along a longitudinal axis of a photosensitive body drum  1302  that is the irradiation target object. 
     The photosensitive body drum  1302  starts to be rotated in a direction of an arrow shown in  FIG. 13A , and is charged at a high electrical potential by a charging device  1305 . The charged portion reaches the scan line of the scanning light beam  233  as the drum  1302  rotates. ON and OFF of the light source  231  (see  FIG. 1 ) are repeated in such a manner that the scanning light beam  233  is applied to each desired position. The electrical potential of the portion irradiated with the scanning light beam  233  emitted from the light source  231  and deflected by the optical deflector is lowered. Thus, an electrostatic latent image is produced. A developing device  1304  develops the portion of the electrostatic latent image by using magnetic toner including a positive-charged black constituent, for example. The developed toner is transferred to a paper by a transfer member  1303 , for example. Thus, the printing process is completed. 
     In this exemplary embodiment, the delay phase difference measured during the driving is caused to be approximately coincident with the target delay phase difference. Accordingly, even in a vibratory system with plural movable bodies, the driving frequency for achieving an efficient driving can be determined. Thus, even when the resonance frequency changes due to a change in the ambient condition such as temperature, the vibratory system can be always driven efficiently in response to the above change. 
     A second exemplary embodiment will be described. In the second exemplary embodiment, the condition for changing the driving frequency is different from that in the first exemplary embodiment. In the first exemplary embodiment, the driving frequency is changed if even a slight difference exists between the delay phase difference measured during the driving and data of the target delay phase difference stored in the storing portion  102 . In contrast, in the second exemplary embodiment, the driving frequency is changed only when the difference reaches a predetermined threshold or more. 
     In the second exemplary embodiment, the predetermined threshold is stored in a storing portion  103  of the controller  100 . In a conditional step (S 115 ) in  FIG. 14 , the controller  100  judges if the difference between the delay phase difference measured during the driving and the target delay phase difference is above the threshold φth or more. If so, the driving frequency is changed, and if not, the printing is continued.  FIG. 15  shows changes in the driving frequency-phase characteristic and the driving frequency in the second exemplary embodiment. In this embodiment, the frequency of changing the driving frequency is reduced. Accordingly, burden on the controlling portion is alleviated. As for the other, the second exemplary embodiment is the same as the first exemplary embodiment. 
     A third exemplary embodiment will be described. In the third exemplary embodiment, limitation is made to the changing timing of the driving frequency in the step (S 116 ) in the first exemplary embodiment. In the image forming apparatus, there are image describing regions  161  and  164  and non-image describing region  162  and  163  as illustrated in  FIGS. 16A and 16B , for example. If the driving frequency is changed when the scanning light beam is present on the image describing region, there is a possibility of appearing adverse influences on the quality of an image formed with the scanning light beam. It is often desirable to change the driving frequency while the scanning light beam is present on the non-image describing region. 
     In the third exemplary embodiment, the controlling portion  250  recognizes the non-image describing region by acquiring image information from an image memory  1700  illustrated in  FIG. 17 .  FIG. 18  shows a flow chart of the third exemplary embodiment. This flow chart is different from the flow chart of  FIG. 14  in that if the scanning light beam is present on the non-image describing region or not is judged (S 301 ) when a change in the driving frequency is needed. When the scanning light beam is present on the image describing region, a present delay phase difference between the drive phase and the vibration phase is again measured. Based on the result of repeated measurement, when a necessity of changing the driving frequency still exists and the scanning light beam is present on the non-image describing region of the irradiation target object, the driving frequency is changed (S 116 ). As for the other, the third exemplary embodiment is the same as the first exemplary embodiment. 
     A fourth exemplary embodiment will be described. In the fourth exemplary embodiment, a vibratory system includes a single movable body.  FIG. 20  shows the relationship between the drive signal and the angular displacement. Each of the drive signal and the angular displacement has a single frequency component only. The phase difference between the drive phase and the vibration phase is φ. 
       FIG. 21  shows a flow chart of this exemplary embodiment. In the driving using one frequency, there is no need of the application of the drive signal component of the integer-fold wave (see S 104  in  FIG. 14 ) and the phase control between two frequency components in the angular displacement (see S 105  in  FIG. 14 ). These steps are necessary in the system including two vibrators. Therefore, respective portions in the controlling portion can be simplified in the fourth exemplary embodiment. As illustrated in the flow chart of  FIG. 21 , the operation principle of this exemplary embodiment is the same as that of the first exemplary embodiment. 
     While the present invention has been described with reference to exemplary embodiments, it is to be understood that the invention is not limited to the disclosed exemplary embodiments. The scope of the following claims is to be accorded the broadest interpretation so as to encompass all such modifications and equivalent structures and functions. 
     This application claims the benefit of Japanese Patent Application Nos. 2008-97394, filed Apr. 3, 2008, and 2009-50013, filed Mar. 4, 2009, which are incorporated by reference herein in their entirety.