Hydroforming is one the most commonly used sheet metal forming methods due to its numerous advantages. These include high surface quality, high dimensional accuracy resulting from spring return, reducing the number of parts production stages, and forming complex shapes. However, hydroforming requires high computational accuracy in die and process design in terms of complexity and high sensitivity to properly control metal flow and prevent defects. The sheet hydroforming process is one of the deep drawing methods. In order to overcome the limitations of the deep drawing process, in addition to applying pressing force, fluid pressure is also used for forming operations [1, 2]. The fluid can replace the punch or matrix in the conventional deep drawing process in the sheet hydroforming process. This process is divided into two general categories, matrix-fluid and punch-fluid .
Due to the disadvantages of the sheet hydroforming process, such as increasing the forming force and low process speed, relatively new processes have been used in combination with it; one of these methods is forming with the assistance of ultrasonic vibration. Ultrasonic vibration are made of sound waves generated by the vibrational motion of matter and are transmitted by the material environment and transfer the energy from the vibration source. These waves are mechanical longitudinal vibrations (with a frequency range of 20 to 100 kHz) and can pass through materials and transfer energy. Therefore, it can also be used in metal forming processes .
The application of ultrasonic vibration to the plasticity behavior of metals began in the 1950s. The first investigation on the effect of ultrasonic vibration on the plastic deformation behavior of metals was conducted by Blaha and Langenecker in 1955 . They tested the strain of single-crystal metal specimens immersed in Tetrachloromethane under the influence of ultrasonic vibration with a wide frequency range (up to 800 kHz). Ultrasonic vibration increased the strain until the moment of fracture and also increased the ultimate tensile strength. They concluded that ultrasonic energy is more effective than thermal energy in reducing tensile stress due to the more excellent absorption of ultrasonic in the dislocations, which increases the plastic flow of the material. Later, the phenomenon of reducing flow stress by applying ultrasonic vibration observed in different metals was known as Blaha or volume effect.
Nevill and Brotzen  suggested that the reduction in flow stress could be easily attributed to the superposition mechanism of stress because although the mean applied stress was reduced, the maximum stress was equal to the yield stress. In 1966, Langenecker investigated the effect of ultrasonic vibrations on the mechanical properties of metals such as zinc and aluminum and showed that in the tensile test, the results could not be explained by acoustic stress alone. He subsequently explained the various possible mechanisms and suggested that the reduction of the yield stress is directly proportional to the ultrasonic energy applied to the sample. The superposition mechanism prevails when the high thermal energy above the sample causes it to soften .
In addition, ultrasonic effects on friction properties were reported as surface effects. For example, Pohlman, E. Lehfeldt  also studied the effects of external friction. By applying vibrations in three directions perpendicular to the slip surface, they found that the friction was reduced in all cases. It is attributed to unwanted ultrasonic shearing due to adhesion of the surface due to local pressure.
In general, the mechanism of effectiveness of ultrasonic vibration in metal forming processes is divided into two categories: surface effects (macroscopic) and volume effects (microscopic) [8–11]. the surface effects of ultrasonic vibration in the metal forming processes are the reduction of sliding friction at the contact surfaces of the die and the workpiece (ultrasonic lubrication) [12, 13] and the reduction of spring back and dimensional and geometric stability of the workpiece . The volume effects of ultrasonic vibration in metal forming processes also include changes in brittleness and plasticity (acoustic softening or hardening property) or the same effect of Blaha and Langenecker , significant reduction of forming force , increasing formability  and changes in microstructure .
Due to these promising results, various researches have been done on ultrasonic vibration in various metal forming processes. Almost all of them with ultrasonic vibration, a significant improvement in the forming process were reported both in reducing the forming force and increasing the apparent formability. It has been reported, and various theories have been proposed for these observations.
For example, in the deep drawing process, which is close to the subject of research in this article, Kristoffy  investigated the effect of ultrasonic vibration on the deep drawing process by applying vibration to punch(axial) and matrix(radial) separately and combinatory on AISI4130 and AL1010. He concluded that applying ultrasonic vibrations has generally reduced the forming force (about 20%). In addition, by an increase of vibrations amplitude, there will be more decrease informing forces.
Then Jima et al.  also studied different methods of applying ultrasonic vibration in the die of deep drawing to improve the process and showed that by applying vibration with frequency 20 and 28 kHz frequency, the limiting drawing ratio (LDR) from 2.68 to 3.01, from 2.58 to 2.94 and from 2.38 to 2.77 for deep-drawing steel sheet (SPCE), cold-rolled steel sheet (SPCC) and stainless steel (SUS304), incased. However, they observed that the application of axial vibration in different parts of the deep drawing die leads to higher LDRs than radial vibration.
For the first time, Huang et al.  investigated the micro-scale deep drawing process with ultrasonic vibration on 304 stainless steel foils in different thicknesses to determine the effect of ultrasonic vibration on the formability and LDR of the specimens. They showed that applying ultrasonic vibration to the matrix increases the LDR for different thicknesses, and the amplitude of oscillation affects the LDR and the quality of the cup.
In a similar case, Shaykholeslami et al.  numerically studied and simulated the finite element method of a cylindrical deep drawing process with a rubber matrix in the presence of ultrasonic vibration and analyzed the effect of amplitude and frequency of ultrasonic vibration on this process. They applied vibration axially to the punch and performed simulations at three frequencies and three amplitudes. As a result of this research, it was found that the forming force is significantly reduced, and also, with increasing the amplitude and frequency, the forming depth increases, and the process stability improves.
Kakinoki et al.  also introduced a method for detecting the amount of Wrinkling in the sheet forming process using ultrasonic vibrations. For this purpose, by applying ultrasonic vibration(on the MHz frequency scale) axially to the point of interface between the die and workpiece in a deep drawing die and measuring its relative reflection intensity, they were able to measure the number of wrinkles during the process and determine parameters affecting it. They concluded that as the wrinkle height increases, the relative reflection intensity decreases, and as the wrinkle wavelength increases, the difference between the maximum and minimum relative reflection intensity increases.
Regarding the subject of the forthcoming research on the application of ultrasonic vibration in hydroforming, Eftekhari Shahri et al.  investigated the improvement of formability in the tube hydroforming process by applying ultrasonic vibration. This process was performed by simulation and experimentally to convert copper tube transversely to square profile. The results showed a significant improvement in the formability of the tube so that while the wall thickness became uniform, the forming rate increased, and the spring back decreased. In this case, Zarei et al.  conducted a study on hydroforming of copper tubes using ultrasonic vibration, in which they simulated the bulging process using the finite element method. This simulation introduced ultrasonic vibration into the tube longitudinally and loaded them with the specified pressure path. This study concluded that the amount of forming force depends on the amplitude of the applied vibration and can be reduced by up to 20%. Also, the application of ultrasonic vibration affects increasing the diameter of the bulge.
Given the above, it can be seen that the effects of ultrasonic vibration on the sheet hydroforming process and the study of the behavior of metal sheet forming in the presence of oil pressure, under the influence of this vibration in none of its states have been studied. Therefore, the purpose of this study is to predict the double efficiency of the hydroforming process with the presence of ultrasonic vibration, using ultrasonic vibration in sheet hydroforming die (radially on the punch), and to investigate its effects so that the benefits of both methods can be used. It increased formability as much as possible.
For this purpose, the effect of ultrasonic vibration on the sheet hydroforming process has been investigated experimentally to determine the effects of applying ultrasonic vibration on the parameters of the sheet hydroforming process compared to the conventional deep drawing. Ordinary and straightforward sheet hydroforming will change. it is necessary to first explain how to apply ultrasonic vibration in sheet hydroforming to the punch in the radial state and then determine the quantitative and qualitative effects of applying ultrasonic vibrations in sheet hydroforming.