Model force-speed characteristics of linear piezoelectric traveling wave motor

Translational Actuators

The Piezo Linear Motor block represents the force-speed characteristics of a linear piezoelectric traveling wave motor. The block represents the force-speed relationship of the motor at a level that is suitable for system-level modeling. To simulate the motor, the block uses the following models:

The motor is unpowered when the physical signal input *v* is
zero. This corresponds to applying zero RMS volts to the motor. In
this scenario, the block models the motor using the following elements:

An mass whose value is the

**Plunger mass**parameter value.A friction whose characteristics you specify using the parameter values in the

**Motor-Off Friction**tab.The block uses a Simscape™ Translational Friction block to model the friction component. For detailed information about the friction model, see the Translational Friction block reference page.

When the motor is active, Piezo Linear Motor block represents the motor characteristics using the following equivalent circuit model.

In the preceding figure:

The AC voltage source represents the block's physical signal input of frequency

*f*and magnitude*v*.The resistor

*R*provides the main electrical and mechanical damping term.The inductor

*L*represents the rotor vibration inertia.The capacitor

*C*represents the piezo crystal stiffness.The capacitor

*C*represents the phase capacitance. This is the electrical capacitance associated with each of the two motor phases._{p}The force constant

*k*relates the RMS current_{f}*i*to the resulting mechanical force.The quadratic mechanical damping term, $$\lambda {\dot{x}}^{2}$$, shapes the force-speed curve predominantly at speeds close to maximum RPM. $$\dot{x}$$ is the linear speed.

The term $$M\dot{x}$$ represents the plunger inertia.

At model initialization, the block calculates the model parameters *R*, *L*, *C*, *k _{t}* and

**Rated force****Rated speed****No-load maximum speed****Maximum (stall) force**

These parameter values are defined for the **Rated
RMS voltage** and **Motor natural frequency** (or
rated frequency) parameter values.

The quadratic mechanical damping term produces a quadratic force-speed curve. Piezoelectric motors force-speed curves can typically be approximated more accurately using a quadratic function than a linear one because the force-speed gradient becomes steeper as the motor approaches the maximum speed.

If the plunger mass *M* is not specified on
the datasheet, you can select a value that provides a good match to
the quoted response time. The response time is often defined as the
time for the rotor to reach maximum speed when starting from rest,
under no-load conditions.

The quality factor that you specify using the **Resonance
quality factor** parameter relates to the equivalent circuit
model parameters as follows:

$$Q=\frac{1}{R}\sqrt{\frac{L}{C}}$$

This term is not usually provided on a datasheet. You can calculate its value by matching the sensitivity of force to driving frequency.

To reverse the motor direction of operation, make the physical
signal input *v* negative.

The block has the following limitations:

When the motor is powered, the model is valid only between zero and maximum speed, for the following reasons:

Datasheets do not provide information for operation outside of normal range.

Piezoelectric motors are not designed to operate in the powered braking and generating regions.

The block behaves as follows outside the valid operating region:

Below zero speed, the model maintains a constant force with a zero speed value. The zero speed value is the

**Maximum (stall) force**parameter value if the RMS input voltage equals the**Rated RMS voltage**parameter value, and the frequency input equals the**Motor natural frequency**parameter value.Above maximum speed, the model produces the negative force predicted by the equivalent circuit model, but limits the absolute value of the force to the zero-speed maximum force.

The force-speed characteristics are most representative when operating the model close to the rated voltage and resonant frequency.

**Motor natural frequency**Frequency at which the piezoelectric crystal naturally resonates. For most applications, set the input signal at port

`f`

to this frequency. To slow down the motor, for example in a closed-loop speed control, use a frequency slightly less than the motor natural frequency. The default value is`92`

kHz.**Rated RMS voltage**Voltage at which the motor is designed to operate. The default value is

`5.7`

V.**Rated force**Force the motor delivers at the rated RMS voltage. The default value is

`0.1`

N.**Rated speed**Motor speed when the motor drives a load at the rated force. The default value is

`50`

mm/s.**No-load maximum speed**Motor speed when driving no load and powered at the rated voltage and driving frequency. The default value is

`150`

mm/s.**Maximum (stall) force**Maximum force the motor delivers when actively driving a load and powered at the rated voltage and frequency. The default value is

`0.15`

N.**Note:**The**Holding force**parameter value, the load force the motor holds when stationary, may be greater than the**Maximum (stall) force**parameter value.**Resonance quality factor**Quality factor

*Q*that specifies how force varies as a function of driving frequency. Increasing the quality factor results in a much more rapid decrease in force as driving frequency is moved away from the natural frequency. The default value is`100`

.**Capacitance per phase**Electrical capacitance associated with each of the two motor phases. The default value is

`5`

nF.

**Plunger mass**Mass of the moving part of the motor. The default value is

`0.3`

g.**Initial rotor speed**Rotor speed at the start of the simulation. The default value is

`0`

mm/s.

**Holding force**The sum of the Coulomb and the static frictions. It must be greater than or equal to the

**Coulomb friction force**parameter value. The default value is`0.3`

N.**Coulomb friction force**The friction that opposes rotation with a constant force at any velocity. The default value is

`0.15`

N.**Viscous friction coefficient**Proportionality coefficient between the friction force and the relative velocity. The parameter value must be greater than or equal to zero. The default value is

`1e-05`

s*N/mm.**Transition approximation coefficient**The parameter sets the coefficient value that is used to approximate the transition between the static and the Coulomb frictions. For detailed information about the coefficient,

*c*, see the Simscape Translational Friction block reference page. The default value is_{v}`0.1`

s/mm.**Linear region velocity threshold**The parameter sets the small vicinity near zero velocity, within which friction force is considered to be linearly proportional to the relative velocity. MathWorks recommends that you use values between

`1e-6`

and`1e-4`

mm/s. The default value is`0.1`

mm/s.

The block has the following ports:

`f`

Physical signal input value specifying the motor driving frequency in Hz.

`v`

Physical signal input magnitude specifying the RMS supply voltage, and sign specifying the direction of rotation. If

`v`

is positive, then a positive force acts from port C to port R.`i`

Physical signal output value that is the RMS phase current.

`vel`

Physical signal output value that is the linear speed of the rotor.

`C`

Mechanical translational conserving port.

`R`

Mechanical translational conserving port.

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